Abstract
The Green–Griffiths–Lang and Kobayashi hyperbolicity conjectures for generic hypersurfaces of polynomial degree are proved using intersection theory for non-reductive geometric invariant theoretic quotients and recent work of Riedl and Yang.
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1 Introduction
The goal of this paper is to apply a recent extension [10, 14] of geometric invariant theory (GIT) to suitable non-reductive actions to study hyperbolicity of generic hypersurfaces in a projective space. We use the results of [14] to construct new compactifications of bundles of invariant jet differentials over complex manifolds. Intersection theory developed in [10] for non-reductive GIT quotients, combined with the strategy of [25], leads us to a proof of the Green–Griffiths–Lang (GGL) conjecture for a generic projective hypersurface whose degree is bounded below by a polynomial in its dimension. A recent result of Riedl and Yang [37] then implies the polynomial Kobayashi conjecture. These results are significant improvements of the earlier known degree bounds, from \((\sqrt{n}\log n)^{n}\) to \(16n^{3}(5n+4)\) in the case of the GGL conjecture and from \((n\log n)^{n}\) to \(16(2n-1)^{3}(10n-1)\) in the case of the Kobayashi conjecture.
A projective variety \(X\) is called Brody hyperbolic if there is no non-constant entire holomorphic curve in \(X\), i.e. any holomorphic map \(f: {\mathbb{C} }\to X\) must be constant. Hyperbolic algebraic varieties have attracted considerable attention, in part because of their conjectured diophantine properties. For instance, Lang [31] has conjectured that any hyperbolic complex projective variety over a number field \(K\) can contain only finitely many rational points over \(K\). In 1970 Kobayashi [30] formulated the following conjecture (more precisely, this is a slightly stronger version of Kobayashi’s original conjecture in which ‘very general’ replaced ‘generic’):
Conjecture 1.1
Kobayashi conjecture, 1970
A generic hypersurface \(X\subseteq {\mathbb{P} }^{n+1}\) of degree \(d_{n}\) is Brody hyperbolic if \(d_{n}\) is sufficiently large.
This conjecture has become a landmark in the field and has been the subject of intense study [2, 3, 16, 17, 22]. For more details on recent results see the survey papers [21, 24]. Siu [41] and Brotbek [16] proved the Kobayashi hyperbolicity of projective hypersurfaces of sufficiently high (but not effective) degree, and effective degree bounds were worked out by Deng [22] and Demailly [21]. The conjectured optimal degree bound is \(d_{1}=4\), \(d_{n}=2n+1\) for \(n=2,3,4\) and \(d_{n}=2n\) for \(n\ge 5\), see [21]. The best known bound before the results of this paper was \(d_{n}=(n \log n)^{n}\) by Merker and The-Anh Ta [34].
A related, but stronger, conjecture is the Green–Griffiths–Lang (GGL) conjecture formulated in 1979 by Green and Griffiths [27] and in 1986 by Lang [31].
Conjecture 1.2
Green-Griffiths-Lang conjecture, 1979
Any projective algebraic variety \(X\) of general type contains a proper algebraic subvariety \(Y\subsetneqq X\) such that every nonconstant entire holomorphic curve \(f:{\mathbb{C} }\to X\) satisfies \(f({\mathbb{C} }) \subseteq Y\).
In particular, a generic projective hypersurface \(X\subseteq {\mathbb{P} }^{n+1}\) is of general type if \(\deg (X)\ge n+3\). A positive answer to the GGL conjecture has been given for surfaces by McQuillan [33] under the assumption that the second Segre number \(c^{2}_{1}-c_{2}\) is positive. Siu [39–42] and Demailly [19] developed a powerful strategy to approach the conjecture for generic hypersurfaces \(X\subseteq {\mathbb{P} }^{n+1}\) of high degree. Following this strategy, combined with techniques of Demailly [19], the first effective lower bound for the degree of a generic hypersurface in the GGL conjecture was given by Diverio, Merker and Rousseau [25], where the conjecture for generic projective hypersurfaces \(X\subseteq {\mathbb{P} }^{n+1}\) of degree \(\deg (X)>2^{n^{5}}\) was confirmed. In [3] the first author introduced equivariant localisation on the Demailly–Semple tower and adapted the argument of [25] to improve this lower bound to \(\deg (X)>n^{8n}\). The residue formula of [3] was later studied and further analyzed by Darondeau [17]. The best bound before the results of this paper for the Green–Griffiths–Lang Conjecture was \(\deg (X)>(\sqrt{n}\log n)^{n}\), due to Merker and The-Anh Ta [34] achieved by a deeper study of the formula of [3].
In this paper we replace the Demailly–Semple bundle with a computationally more efficient algebraic model coming from non-reductive geometric invariant theory [15] and apply the equivariant intersection theory developed in [10] to prove
Theorem 1.3
Polynomial Green-Griffiths-Lang theorem for projective hypersurfaces
Let \(X\subseteq {\mathbb{P} }^{n+1}\) be a generic smooth projective hypersurface of degree \(\deg (X)\ge 16n^{3}(5n+4)\). Then there is a proper algebraic subvariety \(Y\subsetneqq X\) containing all nonconstant entire holomorphic curves in \(X\).
Recently Riedl and Yang [37] proved the following beautiful statement: if there are integers \(d_{n}\) for all positive \(n\) such that the GGL conjecture for generic hypersurfaces of dimension \(n\) holds for degree at least \(d_{n}\) then the Kobayashi conjecture is true for generic hypersurfaces with degree at least \(d_{2n-1}\). Using this, Theorem 1.3 immediately implies
Theorem 1.4
Polynomial Kobayashi theorem
A generic smooth projective hypersurface \(X\subseteq {\mathbb{P} }^{n+1}\) of degree \(\deg (X)\ge 16(2n-1)^{3}(10n-1)\) is Brody hyperbolic.
The strategy of Demailly and Siu is based on first establishing algebraic degeneracy of holomorphic curves \(f:{\mathbb{C} }\to X\), in the sense of proving the existence of certain polynomial differential equations of some order \(k\), and finding enough such equations \(P(f',\ldots ,f^{(k)})=0\) so that they cut out a proper algebraic locus \(Y\subsetneqq X\). The central tool for finding polynomial differential equations is the study of the bundle \(J_{k}X\) of \(k\)-jets of germs of holomorphic curves \(f:{\mathbb{C} }\to X\) over \(X\), and the associated Green–Griffiths bundles \(E_{k}^{GG}=\oplus _{m=0}^{\infty }E_{k,m}^{GG}\) of algebraic differential operators where the fibres of \(E_{k,m}^{GG}\) are polynomial functions \(Q(f',\ldots ,f^{(k)})\) of weighted degree \(m\) in \(f',\ldots ,f^{(k)}\). In [19] Demailly introduced the subbundles \(E_{k,m} \subseteq E_{k,m}^{GG}\) consisting of jet differentials which are (semi-)invariant under reparametrisation of the source ℂ. The group \(\mathrm{Diff}_{k}(1)\) of \(k\)-jets of reparametrisation germs \(({\mathbb{C} },0) \to ({\mathbb{C} },0)\) at the origin acts fibrewise on \(J_{k}X\), and \(\oplus _{m=1}^{\infty }E_{k,m}\) is the graded algebra of jet differentials which are invariant under the maximal unipotent subgroup \(U_{k}\) of \(\mathrm{Diff}_{k}(1)\). This bundle gives a better reflection of the geometry of entire curves in \(X\), since it only depends on the images of such curves and not on their parametrisations. However, it also comes with a technical difficulty: the reparametrisation group \(\mathrm{Diff}_{k}(1)\) is not reductive, and so the classical geometric invariant theory of Mumford [35] cannot be applied to study the invariants and construct a compactification of a quotient of a nonempty open subset \(J_{k}^{\mathrm{reg}}X\) of \(J_{k}X\) by \(\mathrm{Diff}_{k}(1)\) (cf. [26]). Until recently, there existed only two different constructions for the compactification of these quotients.
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1.
In [19] Demailly describes a smooth compactification of \(J_{k}^{\mathrm{reg}}X/\mathrm{Diff}_{k}(1)\) as a tower of projectivised bundles on \(X\) — the Demailly–Semple bundle — endowed with line bundles such that sections of their powers give \(\mathrm{Diff}_{k}(1)\)-(semi)-invariants. Global sections of properly chosen twisted tautological line bundles over the Demailly–Semple bundle give algebraic differential equations of degree \(k\). This model was extensively and successfully used in the past few decades, and it has a vast literature in hyperbolicity questions (see also [21, 24]). The main numerical breakthrough in the Green-Griffiths-Lang conjecture using the Demailly-Semple tower was achieved in [25], where the first effective bound for the degree of a generic projective hypersurface was calculated. However, as was pointed out in [3], we cannot expect better than an exponential bound in the GGL conjecture using the Demailly-Semple model.
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2.
In [2] the first author shows that the curvilinear component of the punctual Hilbert scheme of \(k\) points on \({\mathbb{C} }^{n}\) provides natural compactifications of the fibres of \(J_{k}^{\mathrm{reg}}X/\mathrm{Diff}_{k}(1)\) over \(X\). Sections of the tautological bundle give invariant jet differentials, and equivariant localisation developed in [1, 5–7, 12, 13] gives information on the intersection theory of this curvilinear component. In [2] it is shown that the GGL conjecture for generic hypersurfaces with polynomial degree follows from a classical positivity conjecture of Rimányi [38] for Thom polynomials. However, Rimányi’s conjecture is currently out of reach.
The first key idea of this paper is to replace these existing models with a new construction, coming from a recently developed extension of geometric invariant theory (GIT) to suitable non-reductive group actions [14]. We construct a projective completion \(\mathcal{X}_{k}^{\mathrm{GIT}}\) of \(\mathcal{X}_{k}^{\mathrm{reg}} = {J}_{k}^{\mathrm{reg}} X/ \mathrm{Diff}_{k}(1)\), which fibres over \(X\) with fibres given by non-reductive GIT quotients. The projective completion \(\mathcal{X}_{k}^{\mathrm{GIT}}\) is endowed with a relatively ample line bundle such that sections of its powers give (semi-)invariant jet differentials. From this we construct a tautological line bundle over \(\mathcal{X}_{k}^{\mathrm{GIT}}\), and following Diverio–Merker–Rousseau [25] (with the improved pole order obtained by Darondeau [18] for slanted vector fields) and using the holomorphic Morse inequalities [20, 43], we show that the existence of nonzero sections of this line bundle follows from the positivity of a certain tautological integral over \(\mathcal{X}_{k}^{\mathrm{GIT}}\).
The second key ingredient of this paper is the cohomological intersection theory for non-reductive GIT quotients developed in [10], which allows us to prove the positivity of this integral at the critical order \(k=n\) for hypersurfaces with polynomial degree.
The layout of the paper is as follows. Background material needed on jet differentials and on non-reductive GIT is recalled in \(\S \)2 and \(\S \)3. In \(\S \)4 we summarise the results of [10] on moment maps for nonreductive quotients and their cohomological applications. \(\S \)5 contains the construction of the projective completion \(\mathcal{X}_{k}^{\mathrm{GIT}} \) of \({J}_{k}^{\mathrm{reg}} X/\mathrm{Diff}_{k}(1)\) fibring over \(X\), and describes the fibres \(X_{k}^{\mathrm{GIT}}\) of \(\mathcal{X}_{k}^{\mathrm{GIT}} \to X\) as non-reductive GIT quotients, with explicit descriptions when \(k=2,3\). In \(\S \)6 it is shown that the proof of Theorem 1.3 reduces to proving that a tautological integral
is strictly positive for suitable \(\delta >0\) where \(n = \dim X\). In \(\S \)7 the cohomology of the fibres \(X_{k}^{\mathrm{GIT}}\) is studied, and \(\S \)8 derives integration formulas on \(\mathcal{X}_{k}^{\mathrm{GIT}}\). In \(\S \)9 the integral \(\int _{\mathcal{X}_{n}^{\mathrm{GIT}}} I_{n,\delta} \) is calculated explicitly when \(n=2\), and then the proof of Theorem 1.3 is completed by demonstrating positivity for general \(n\).
The authors would like to thank the anonymous referee for helpful comments on an earlier version of this paper. The first author was partially supported by AUFF Starting Grant 29289.
2 Jet differentials
The central object of this paper is the algebra of (semi-)invariant jet differentials under reparametrisation of the source space ℂ. For more details see the survey papers of Demailly [19] and Diverio–Rousseau [24].
2.1 Jets of holomorphic maps
If \(u\), \(v\) are positive integers let \(J_{k}(u,v)\) denote the vector space of \(k\)-jets of holomorphic maps \(({\mathbb{C} }^{u},0) \to ({\mathbb{C} }^{v},0)\) at the origin; that is, the set of equivalence classes of maps \(f:({\mathbb{C} }^{u},0) \to ({\mathbb{C} }^{v},0)\), where \(f\sim g\) if and only if \(f^{(j)}(0)=g^{(j)}(0)\) for all \(j=1,\ldots ,k\). This is a finite-dimensional complex vector space, which can be identified with \(J_{k}(u,1) \otimes {\mathbb{C} }^{v}\); hence \(\dim J_{k}(u,v) =v \binom{u+k}{k}-v\). We will call the elements of \(J_{k}(u,v)\) map-jets of order \(k\), or simply map-jets.
Eliminating the terms of degree \(k+1\) results in a surjective algebra homomorphism \(J_{k}(u,1) \twoheadrightarrow J_{k-1}(u,1)\), and the sequence of such surjections \(J_{k}(u,1) \twoheadrightarrow J_{k-1}(u,1) \twoheadrightarrow \cdots \twoheadrightarrow J_{1}(u,1)\) induces an increasing filtration of \(J_{k}(u,1)^{*}\):
Using the standard coordinates on \({\mathbb{C} }^{u}\) and \({\mathbb{C} }^{v}\), a \(k\)-jet \(f \in J_{k}(u,v)\) can be identified with its collection of derivatives at the origin, the vector \((f'(0),f''(0),\ldots , f^{(k)}(0))\), where \(f^{(j)}(0)\in \mathrm{Hom}(\mathrm{Sym}^{j}{\mathbb{C} }^{u},{ \mathbb{C} }^{v})\). Here \(\mathrm{Sym}^{l}\) denotes the symmetric tensor product. In this way we get an isomorphism
Map-jets can be composed via substitution and elimination of terms of degree greater than \(k\), leading to the composition map
When \(k=1\), we can identify \(J_{1}(u,v)\) with the space of \(u\)-by-\(v\) matrices, and (2.3) reduces to multiplication of matrices.
We will call a jet \(\gamma \in J_{k}(u,v)\) regular if \(\gamma '(0)\) is has maximal rank, and we will use the notation \(J_{k}^{\mathrm{reg}}({u},{v})\) for the set of regular maps. When \(u=v\) we get a group
which we will call the \(k\)-jet diffeomorphism group.
Remark 2.1
Note that from (2.3) we obtain commuting right and left actions of \(\mathrm{Diff}_{k}(1)\) and \(\mathrm{Diff}_{k}(n)\) on \(J_{k}(1,n)\).
2.2 Jet bundles and differential operators
Let \(X\) be a smooth projective variety of dimension \(n\). Following Green and Griffiths [27] we let \(J_{k}X \to X\) be the bundle of \(k\)-jets of germs of parametrised curves in \(X\); that is, \(J_{k}X\) is the of equivalence classes of germs of holomorphic maps \(f:({\mathbb{C} },0) \to (X,p)\), where the equivalence relation ∼ is such that \(f\sim g\) if and only if the derivatives \(f^{(j)}(0)\) and \(g^{(j)}(0)\) are equal for \(0\le j \le k\) when computed in some local holomorphic coordinate system on an open neighbourhood of \(p\in X\). The projection map \(J_{k}X \to X\) is given by \(f \mapsto f(0)\), and the elements of the fibre \(J_{k}X_{p}\) can be represented by Taylor expansions
up to order \(k\) at \(t=0\) of \({\mathbb{C} }^{n}\)-valued maps \(f=(f_{1},f_{2},\ldots , f_{n})\) on open neighbourhoods of 0 in ℂ. Locally in these coordinates elements of the fibre \(J_{k}X_{p}\) can be identified with \(k\)-tuples of vectors \((f'(0),\ldots , f^{(k)}(0)/k!) \in ({\mathbb{C} }^{n})^{k}\), so the fibre can be identified with \(J_{k}(1,n)\).
Note that \(J_{k}X\) is not a vector bundle over \(X\) since the transition functions are polynomial but not linear, see §5 of Demailly [19]. In fact, let \(\mathrm{Diff}_{X}\) denote the principal \(\mathrm{Diff}_{k}(n)\)-bundle over \(X\) formed by all local polynomial coordinate systems on \(X\). Then we have an identification
of \(J_{k}X\) with the associated affine bundle whose structure group is \(\mathrm{Diff}_{k}(n)\), acting on \(J_{k}(1,n)\) as in Remark 2.1. With respect to this identification, \(\mathrm{Diff}_{k}(1)\) acts on \(J_{k}X\) fibrewise via its action on \(J_{k}(1,n)\) as in Remark 2.1.
Let \(J_{k}^{\mathrm{reg}}X\) denote the bundle of \(k\)-jets of germs of parametrised curves \(f:{\mathbb{C} }\to X\) in \(X\) which are regular in the sense that they have nonzero first derivative \(f'(0)\neq 0\). After fixing local coordinates near \(p\in X\), the fibre \(J_{k}^{\mathrm{reg}}X_{p}\) can be identified with \(J_{k}^{\mathrm{reg}}({1},{n})\) and
2.3 Invariant jet differentials
Let \(X\) be a complex \(n\)-dimensional manifold and let \(k\) be a positive integer. Recall that after choosing local coordinates on \(X\) near \(p\) we can identify the fibre \(J^{\mathrm{reg}}_{k}X_{p}\) of \(J^{\mathrm{reg}}_{k} X\) at \(p\) with \(J_{k}^{\mathrm{reg}}({1},{n})\). We can explicitly write out the reparametrisation action (defined at (2.3)) of \(\mathrm{Diff}_{k}(1)\) on \(J_{k}^{\mathrm{reg}}({1},{n})\) as follows. Let \(z f'(0)+\frac{z^{2}}{2!}f''(0)+\cdots +\frac{z^{k}}{k!}f^{(k)}(0) \in J_{k}^{\mathrm{reg}}({1},{n})\) be the \(k\)-jet of a germ at the origin (i.e. no constant term) in \({\mathbb{C} }^{n}\) with \(f^{(i)}(0)\in {\mathbb{C} }^{n}\) for \(1 \leq i \leq k\) such that \(f'(0) \neq 0\), and let \(\varphi (z)=\alpha _{1}z+\alpha _{2}z^{2}+\cdots +\alpha _{k} z^{k} \in J_{k}^{\mathrm{reg}}({1},{1})\) with \(\alpha _{i}\in {\mathbb{C} }\), \(\alpha _{1}\neq 0\). Then the \(k\)-jet of \(f \circ \varphi (z)\) is
where the \((i,j)\) entry of this matrix is \(p_{i,j}(\bar{\alpha})=\sum _{a_{1}+a_{2}+\cdots +a_{i}=j}\alpha _{a_{1}} \alpha _{a_{2}} \ldots \alpha _{a_{i}}\).
Remark 2.2
The linear representation of \(\mathrm{Diff}_{k}(1)\) on \(J_{k}^{\mathrm{reg}}({1},{n})\) given by (2.4) embeds \(\mathrm{Diff}_{k}(1)\) as a upper triangular subgroup of \(\mathrm{GL}(n)\). Thus \(\mathrm{Diff}_{k}(1)\) is a linear algebraic group but is not reductive for \(k \geq 2\), so Mumford’s classical GIT cannot be used to construct compactifications of the orbit space \(J_{k}^{\mathrm{reg}}({1},{n})/ \mathrm{Diff}_{k}(1)\) (cf. [14, 26]).
This matrix group is parametrised along its first row with free parameters \(\alpha _{1}\in {\mathbb{C} }^{*}, \alpha _{2},\ldots , \alpha _{k} \in {\mathbb{C} }\), while the other entries are certain (weighted homogeneous) polynomials in these free parameters. It is a semidirect product
of its unipotent radical \(U_{k}\) by a one-parameter subgroup \({\mathbb{C} }^{*}\) acting diagonally. Here \(U_{k}\) is the subgroup given by substituting \(\alpha _{1}=1\), and the diagonal subgroup \({\mathbb{C} }^{*}\) acts with strictly positive weights \(1\ldots , n-1\) on the Lie algebra \(\mathrm{Lie}(U_{k})\) of \(U_{k}\). In [10, 14] actions of non-reductive groups of this type are studied in a more general context.
The action of \(\lambda \in {\mathbb{C} }^{*}\) on \(k\)-jets is thus described by
Let \(\mathcal{E}_{k,m}\) denote the space of complex-valued polynomials \(Q(f'(0),f''(0),\ldots , f^{(k)}(0))\) on \(J_{k}(1,n)\) of weighted degree \(m\) with respect to this \({\mathbb{C} }^{*}\) action; that is, they satisfy
In [19] Demailly introduced the Green-Griffiths bundle \(E_{k,m}^{GG}\) over \(X\) whose fibres are \(\mathcal{E}_{k,m}\), observing that the concept of polynomial on the fibres of \(J_{k}X\) using local coordinates on \(X\) is well defined. From our viewpoint this can be written as the associated bundle
The Green-Griffiths bundle of order \(k\) is then \(E_{k}^{GG}=\oplus _{m\ge 0} E_{k,m}^{GG}\).
The fibrewise \(\mathrm{Diff}_{k}(1)\) action on \(J_{k}X\) induces an action on \(E_{k,m}^{GG}\). Demailly in [19] defined the bundle of invariant jet differentials of order \(k\) and weighted degree \(m\) as the subbundle \(E_{k,m}\subseteq E_{k,m}^{GG}\) of polynomial differential operators \(Q(f'(0),\ldots , f^{(k)}(0))\) which are invariant under \(U_{k}\); that is for any \(\varphi \in \mathrm{Diff}_{k}(1)\)
We call \(E_{k}=\oplus _{m} E_{k,m}=(\oplus _{m}E_{k,m}^{GG})^{U_{k}}\) the Demailly–Semple bundle of invariant jet differentials.
2.4 Compactifications of \(J_{k}X/\mathrm{Diff}_{k}(1)\)
In order to find and describe invariant jet differentials we can try to construct projective completions of the quasi-projective fibrewise quotient
This quotient fibres over \(X\) (as \(\mathrm{Diff}_{k}(1)\) acts fibrewise) and we can hope to detect invariant jet differentials as global sections of powers of relatively ample line bundles on suitable fibrewise projective completions \(\overline{J_{k}^{\mathrm{reg}}X/\mathrm{Diff}_{k}(1)}\). Indeed this strategy works, and there exist two constructions in the literature.
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1.
The Demailly–Semple tower The first construction goes back to Semple, and was studied and introduced into the study of hyperbolicity questions by the landmark paper of Demailly [19]. The Demailly–Semple tower \(X_{k}^{\mathrm{DS}}\) is an iterated projective bundle over \(X\)
$$ X_{k}^{\mathrm{DS}} \to X_{k-1}^{\mathrm{DS}}\to \cdots \to X_{1}^{ \mathrm{DS}}\to X_{0}^{\mathrm{DS}}=X $$endowed with projections \(\pi _{i,k}:X_{k}^{\mathrm{DS}}\to X_{i}^{\mathrm{DS}}\) and canonical line bundles \(\pi _{i,k}^{*}\mathcal{O}_{X_{i}^{\mathrm{DS}}}(1) \to X_{k}^{ \mathrm{DS}}\) whose sections are global invariant jet differentials. The total space \(X_{k}\) is smooth of dimension \(\dim (X_{k})=n+k(n-1)\). For the details of the construction see [19, 25]. In [3] equivariant localisation was introduced on the Demailly-Semple tower, and following the strategy of [25], the Green–Griffiths–Lang conjecture for generic hypersurfaces with degree at least \(n^{6n}\) was proved. In [3] it was also proved that we cannot expect better than an exponential degree bound with this approach.
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2.
The curvilinear component of the Hilbert scheme of \(k\) points on \({\mathbb{C} }^{n}\) In [1] the first author proves that the curvilinear component \(\mathrm{CHilb}^{k+1}_{0}({\mathbb{C} }^{n})\) of the punctual Hilbert scheme \(\mathrm{Hilb}^{k}_{0}({\mathbb{C} }^{n})\) supported at the origin is a compactification of the fibre \(J_{k}^{\mathrm{reg}}(1,n)/\mathrm{Diff}_{k}(1)\) of \(J_{k}^{\mathrm{reg}}X/\mathrm{Diff}_{k}(1)\). More precisely, in [1] (following [12]) we describe a \(\mathrm{Diff}_{k}(1)\)-invariant map
$$ \phi : J_{k}^{\mathrm{reg}}(1,n) \to \mathrm{Grass}(k,J_{k}(n,1)^{*}). $$(2.6)and we identify the image in the Grassmannian with the punctual curvilinear locus of the Hilbert scheme of \(k+1\) points in \({\mathbb{C} }^{n}\) supported at the origin. The curvilinear locus consists of subschemes of length \(k+1\) on \({\mathbb{C} }^{n}\) which are supported at the origin and are limits of \(k+1\) distinct points colliding along a smooth curve:
$$ {\mathrm{Im}} \,(\phi )=\{I \subset {\mathbb{C} }[x_{1},\ldots , x_{n}]: { \mathbb{C} }[x_{1},\ldots , x_{n}]/I \simeq {\mathbb{C} }[t]/t^{k}\} \subset \mathrm{Hilb}^{k+1}_{0}({\mathbb{C} }^{n}) $$(2.7)The closure of the curvilinear locus is a component of the punctual Hilbert scheme, the so-called curvilinear component \(\mathrm{CHilb}^{k+1}_{0}({\mathbb{C} }^{n})\). Since \(\phi \) is injective on the \(\mathrm{Diff}_{k}(1)\)-orbits,
$$ \mathrm{CHilb}^{k+1}_{0}({\mathbb{C} }^{n})= \overline{ \,{\mathrm{Im}} \,(\phi )}= \overline{J_{k}^{\mathrm{reg}}(1,n)/\mathrm{Diff}_{k}(1)} $$is a projective completion of \(J_{k}^{\mathrm{reg}}(1,n)/\mathrm{Diff}_{k}(1)\). By moving the support on \(X\) we get the fiberwise projective completion
$$ \mathrm{CHilb}^{k+1}(X)= \overline{J_{k}^{\mathrm{reg}}X/\mathrm{Diff}_{k}(1)} $$Using equivariant localisation, in [2] we connect hyperbolicity of hypersurfaces with global singularity theory and Thom polynomials of \(A_{n}\)-singularities. Modulo a positivity conjecture of Rimányi, [2] obtains the Green–Griffiths–Lang conjecture for hypersurfaces with polynomial (\(>n^{8}\)) degree. However, a complete proof of the positivity conjecture currently seems to be out of reach.
This paper introduces a third compactification coming from the recent development of a version of GIT which applies to suitable non-reductive group actions [14]. As we have seen, the reparametrisation group \(\mathrm{Diff}_{k}(1)\) is not reductive, but it is a linear algebraic group with internally graded unipotent radical in the sense of [14], and hence the construction and results of this paper apply. We use the fibrewise completion
where \({\tilde{\mathbb{P} }}[{\mathbb{C} }\oplus J_{k}(1,n)]/\!/ \mathrm{Diff}_{k}(1)\) is a non-reductive GIT quotient of a projective space blown up at the point \([1:0: \cdots :0]\). Care is needed in this construction, since \(J_{k}X\) is an affine bundle but not a vector bundle over \(X\), and the fibrewise action of \(\mathrm{Diff}_{k}(n)\) on \(J_{k}X\) does not extend to the projective space \({\mathbb{P} }[{\mathbb{C} }\oplus \,{\mathrm{Hom}} \,({\mathbb{C} }^{k},{ \mathbb{C} }^{n})]\) nor to its blow-up \({\tilde{\mathbb{P} }}[{\mathbb{C} }\oplus \,{\mathrm{Hom}} \,({\mathbb{C} }^{k},{ \mathbb{C} }^{n})]\). However we will see that the projective non-reductive GIT quotient \({\tilde{\mathbb{P} }}[{\mathbb{C} }\oplus \,{\mathrm{Hom}} \,({\mathbb{C} }^{k},{ \mathbb{C} }^{n})]/\!/ \mathrm{Diff}_{k}(1)\) is a geometric quotient of a \(\mathrm{Diff}_{k}(n)\)-invariant open subset of the blow-up of \(J_{k}(1,n)\) at 0, and thus \(\mathrm{Diff}_{k}(n)\) does act naturally on \({\tilde{\mathbb{P} }}[{\mathbb{C} }\oplus \,{\mathrm{Hom}} \,({\mathbb{C} }^{k},{ \mathbb{C} }^{n})]/\!/\mathrm{Diff}_{k}(1)\) by Remark 2.1. Moreover the affine bundle \(J_{k}X\) over \(X\) has a well-defined zero section which is invariant under the fibrewise action of \(\mathrm{Diff}_{k}(n)\), and we can identify \(\mathcal{X}_{k}^{\mathrm{GIT}}\) with the geometric quotient of the action of \(\mathrm{Diff}_{k}(1)\) on an open subset of the blow-up of \(J_{k}X\) along its zero section.
This means that we can apply the intersection theory and integration formulas for non-reductive GIT quotients proved in [10] as our main computational tool, to integrate over the fibres of \(\mathcal{X}_{k}^{\mathrm{GIT}} \to X\).
Remark 2.3
The Demailly-Semple tower \(X_{k}^{\mathrm{DS}}\), the curvilinear Hilbert scheme \(\mathrm{CHilb}^{k+1}(X)\) and the non-reductive GIT quotient \(\mathcal{X}_{k}^{\mathrm{GIT}}\) are three different fiberwise compactifications of the invariant jet differentials bundle \(J_{k}^{\mathrm{reg}}X/\mathrm{Diff}_{k}(1)\). Description of their birationality is, however, a difficult problem. There are rational maps
where \(\phi \) is defined in (2.6), and \(\psi \) comes from choosing local coordinates on the Demailly-Semple tower, see [8].
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1.
In [4] a toric blow-up of \(\phi \) is defined: we obtain a morphism \(\hat{\phi}: \hat{\mathcal{X}}_{k}^{\mathrm{GIT}} \to \mathrm{CHilb}^{k+1}(X)\) and we pull-back integration over \(\mathrm{CHilb}^{k+1}(X)\) to the non-reductive GIT quotient \(\hat{\mathcal{X}}_{k}^{\mathrm{GIT}}=\widehat{J_{k}X}/\mathrm{Diff}_{k}(1)\) to arrive new toric residue formulas for Thom polynomials of singularities.
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2.
In the unpublished note [8] we study \(\psi \), and the blow-up of the Demailly-Semple bundle at the Wronskian ideal sheaf which defines the indeterminacy locus of \(\psi \). The main goal of [8] was the study of the disribution of torus fixed points (i.e monomial ideals) on the components of the punctual Hilbert scheme. This problem was later solved in [11].
3 Non-reductive geometric invariant theory
In [14] an extension of Mumford’s classical GIT is developed for linear actions of a non-reductive linear algebraic group with internally graded unipotent radical over an algebraically closed field \({\mathbf {k} }\) of characteristic 0; the results are summarised and slightly extended in [10]. In this section we will follow the definitions of [10] and quote its results.
Definition 3.1
We say that a linear algebraic group \(H = U \rtimes R\) has internally graded unipotent radical \(U\) if there is a central one-parameter subgroup \(\lambda :{\mathbb{G} }_{m} \to Z(R)\) of the Levi subgroup \(R\) of \(H\) such that the adjoint action of \({\mathbb{G} }_{m}\) on the Lie algebra of \(U\) has all its weights strictly positive. Then \(\hat{U} = U \rtimes \lambda ({\mathbb{G} }_{m})\) is a normal subgroup of \(H\) and \(H/\hat{U}\cong R/\lambda ({\mathbb{G} }_{m})\) is reductive.
Let \(H=U \rtimes R\) be a linear algebraic group with internally graded unipotent radical \(U\) acting linearly with respect to an ample line bundle \(L\) on a projective variety \(X\); that is, the action of \(H\) on \(X\) lifts to an action on \(L\) via automorphisms of the line bundle. When \(H=R\) is reductive, using Mumford’s classical geometric invariant theory (GIT) [35], we can define \(H\)-invariant open subsets \(X^{s} \subseteq X^{ss}\) of \(X\) (the stable and semistable loci for the linearisation) with a geometric quotient \(X^{s}/H\) and (if \(X^{s}\) is non-empty) projective completion \(X/\!/H \supseteq X^{s}/H\) which is the projective variety associated to the algebra of invariants \(\bigoplus _{k \geq 0} H^{0}(X,L^{\otimes k})^{H}\). Here the variety \(X/\!/H\) is the image of a surjective morphism \(\phi \) from the open subset \(X^{ss}\) of \(X\) such that if \(x,y \in X^{ss}\) then \(\phi (x) = \phi (y)\) if and only if the closures of the \(H\)-orbits of \(x\) and \(y\) meet in \(X^{ss}\). Furthermore the subsets \(X^{s}\) and \(X^{ss}\) can be described using the Hilbert–Mumford criteria for stability and semistability.
Mumford’s GIT does not have an immediate extension to actions of non-reductive linear algebraic groups \(H\), since the algebra of invariants \(\bigoplus _{k \geq 0} H^{0}(X,L^{\otimes k})^{H}\) is not necessarily finitely generated as a graded algebra when \(H\) is not reductive. It is still possible to define semistable and stable subsets \(X^{ss}\) and \(X^{s}\), with a geometric quotient \(X^{s}/H\) which is an open subset of a so-called enveloping quotient \(X\! \mathbin {\text{\rotatebox [origin=c]{70}{\scalebox {1.2}{$\approx $}}}} \!H\) with an \(H\)-invariant morphism \(\phi : X^{ss} \to X\! \mathbin {\text{\rotatebox [origin=c]{70}{\scalebox {1.2}{$\approx $}}}} \!H\), and if the algebra of invariants \(\bigoplus _{k \geq 0} H^{0}(X,L^{\otimes k})^{H}\) is finitely generated then \(X\! \mathbin {\text{\rotatebox [origin=c]{70}{\scalebox {1.2}{$\approx $}}}} \!H\) is the associated projective variety [14, 26]. But in general the enveloping quotient \(X\! \mathbin {\text{\rotatebox [origin=c]{70}{\scalebox {1.2}{$\approx $}}}} \!H\) is not necessarily projective, the morphism \(\phi \) is not necessarily surjective (and its image may be only a constructible subset, not a subvariety, of \(X\! \mathbin {\text{\rotatebox [origin=c]{70}{\scalebox {1.2}{$\approx $}}}} \!H\)). In addition there are in general no obvious analogues of the Hilbert–Mumford criteria.
However when \(H = U \rtimes R\) has internally graded unipotent radical \(U\) and acts linearly on a projective variety \(X\), then provided that we are willing to modify the linearisation of the action by replacing the line bundle \(L\) by a sufficiently divisible tensor power and multiplying by a suitable character of \(H\) (which will not change the action of \(H\) on \(X\)), many of the key features of classical GIT still apply.
Let such an \(H\) act linearly on an irreducible projective variety \(X\) with respect to a very ample line bundle \(L\). Let \(\chi : H \to {\mathbb{G} }_{m}\) be a character of \(H\). Its kernel contains \(U\), and its restriction to \(\hat{U}\) can be identified with an integer so that the integer 1 corresponds to the character of \(\hat{U}\) which fits into the exact sequence \(U \hookrightarrow \hat{U} \to \lambda ({\mathbb{G} }_{m})\). Let \(\omega _{\min}\) be the minimal weight for the \(\lambda ({\mathbb{G} }_{m})\)-action on \(V:=H^{0}(X,L)^{*}\) and let \(V_{\min}\) be the weight space of weight \(\omega _{\min}\) in \(V\). Suppose that \(\omega _{\min}=\omega _{0} < \omega _{1} < \cdots < \omega _{\max} \) are the weights with which the one-parameter subgroup \(\lambda : {\mathbb{G} }_{m} \leq \hat{U}\leq H\) acts on the fibres of the tautological line bundle \(\mathcal{O}_{ {\mathbb{P} }((H^{0}(X,L)^{*})}(-1)\) over points of the connected components of the fixed point set \({\mathbb{P} }((H^{0}(X,L)^{*})^{{\mathbb{G} }_{m}}\) for the action of \({\mathbb{G} }_{m}\) on \({\mathbb{P} }((H^{0}(X,L)^{*})\); since \(L\) is very ample \(X\) embeds in \({\mathbb{P} }((H^{0}(X,L)^{*})\) and the line bundle \(L\) extends to the dual \(\mathcal{O}_{ {\mathbb{P} }((H^{0}(X,L)^{*})}(1)\) of the tautological line bundle on \({\mathbb{P} }((H^{0}(X,L)^{*})\). Note that we can assume that there exist at least two distinct such weights since otherwise the action of the unipotent radical \(U\) of \(H\) on \(X\) is trivial, and so the action of \(H\) is via an action of the reductive group \(R=H/U\).
The linearisation of the action of \(H\) on \(X\) with respect to the ample line bundle \(L^{\otimes c}\) can be twisted by the character \(\chi \) so that the weights \(\omega _{j}\) are replaced with \(\omega _{j}c-\chi \); let \(L_{\chi}^{\otimes c}\) denote this twisted linearisation.
Definition 3.2
If
where \(\epsilon >0\) is sufficiently small, we will call the rational character \(\chi /c\) well adapted to the linear action of \(\hat{U}\). We will call a linearisation well adapted if \(\omega _{\min} <0<\omega _{\min} + \epsilon \) for sufficiently small \(\epsilon >0\). Note that \(\chi /c\) is well-adapted if and only if the linearisation \(L_{\chi}^{\otimes c}\) is well-adapted, and every linearisation \(L\) with at least two different \(\lambda ({\mathbb{G} }_{m})\)-weights has well-adapted rational twists.
Remark 3.3
How small \(\epsilon >0\) should be depends on the situation; see [10] Definition 4.19 and the discussion following it. We certainly require that \(\epsilon \le \omega _{1} - \omega _{\min}\) so that \((\chi |_{\hat{U}})/c \in (\omega _{\min},\omega _{1})\), but we may well require \(\epsilon \) to be smaller than this (cf. the proof of Theorem 4.28 in [10]). The main requirement is that \(\epsilon >0\) should be sufficiently small to ensure via the theory of variation of GIT that, under suitable additional hypotheses (see Definition 3.5 below), twisting the linearisation of the \(H\)-action on \(X\) by the rational character \(\chi /c\) results in (semi)stable loci for reductive subgroups of \(H\) which are independent of \(\chi /c\) subject to the requirement that \(\omega _{\min} < (\chi |_{\hat{U}})/c < \omega _{\min} + \epsilon \). By the Hilbert–Mumford criteria, this can be expressed in terms of the geometry of convex hulls of sets of weights for the action on \(X\) of a maximal torus of \(R\).
Suppose that \(H = U \rtimes R\), with grading one-parameter subgroup \(\lambda :{\mathbb{G} }_{m} \to Z(R)\), acts linearly on a projective variety \(X\) with respect to an ample line bundle \(L\). Let
and
Then \(p: X_{\min}^{0} \to Z_{\min}(X)\) is \(U\)-invariant and \(R\)-equivariant ([10] Lemma 4.16); that is,
Suppose that \(L_{\chi}^{\otimes c}\) is well adapted and let \(X^{s,{\mathbb{G} }_{m}}_{\min +}\) denote the stable subset of \(X\) for the linear action of \(\lambda :{\mathbb{G} }_{m} \to Z(R)\) with respect to \(L_{\chi}^{\otimes c}\). By the Hilbert–Mumford criteria ([35, 36])
indeed \(X^{s,{\mathbb{G} }_{m}}_{\min +}\) is the stable set for the action of \(\lambda :{\mathbb{G} }_{m} \to Z(R)\) with respect to any \(L_{\chi}^{\otimes c}\) such that \(\omega _{\min} < \chi /c < \omega _{1}\). Since the infinitesimal action of a weight vector \(\xi \in {\mathfrak {u}}\) with weight \(\chi \) for the adjoint action of \(\lambda ({\mathbb{G} }_{m})\) on the Lie algebra \({\mathfrak {u}}\) of \(U\) takes a weight vector in \(V\) with weight \(\omega \) to one of weight \(\omega + \chi \), and takes a weight vector in the dual of \(V\) with weight \(\omega \) to one of weight \(\omega - \chi \), the action of \(U\) on \(X\) preserves \(X_{\min}^{0}\), although it does not in general preserve \(Z_{\min}\). Thus if \(u \in U\) we have
Definition 3.4
We say that ‘semistability coincides with stability’ for a well adapted linear action of \(\hat{U}\) if
Note that (∗) holds if and only if we have \(\operatorname{Stab}_{U}(x) = \{e\}\) for all \(x \in X^{0}_{\min}\).
When (∗) holds for a well adapted action of \(\hat{U}\), the min-stable locus for the \(\hat{U}\)-action is
Here the last equation follows from (3.2).
Definition 3.5
A well-adapted linear action on an irreducible projective variety \(X\) is given by the following data:
-
1.
a linear algebraic group \(H = U \rtimes R\) with unipotent radical \(U\) and Levi subgroup \(R\), and a central one-parameter subgroup \(\lambda :{\mathbb{G} }_{m} \to Z(R)\) of \(R\) which grades \(U\) in the sense of Definition 3.1, with a ‘complementary’ connected subgroup \(Z^{\perp}\) of \(Z(R)\) such that \(\mathrm{Lie}Z(R) = \mathrm{Lie}Z^{\perp }\oplus \mathrm{Lie}\lambda ({ \mathbb{G} }_{m})\);
-
2.
a linear action of \(H\) on \(X\) with respect to an ample line bundle \(L\):
-
3.
a character \(\chi : H \to {\mathbb{G} }_{m}\) of \(H\) whose restriction to \(Z^{\perp}\) is zero and whose restriction to \(\hat{U}= U \rtimes \lambda ({\mathbb{G} }_{m})\) is nonzero, and a strictly positive integer \(c\) such that the rational character \(\chi /c\) satisfies
$$ (\chi |_{\hat{U}})/c = \omega _{\min} + \epsilon \text{ where }0< \epsilon < \!< 1. $$
One way to choose \(Z^{\perp}\) is so that its Lie algebra is the orthogonal complement to \(\mathrm{Lie}\lambda ({\mathbb{G} }_{m})\) in \(\mathrm{Lie}Z(R)\) with respect to a suitable inner product. \(Z^{\perp}\) determines and is determined by the ray in the direction of \(\chi \) (that is, consisting of positive scalar multiples of \(\chi \)); the rational character \(\chi /c\) is required to lie on the line generated by this ray, just beyond the point labelled by \(\omega _{\min}\).
Theorem 3.6
[10] Theorem 8
Let \((X,L,H,\hat{U},\chi )\) be a well-adapted linear action satisfying condition (∗) in Definition 3.4. Then
-
1.
the algebras of invariants
$$ \oplus _{m=0}^{\infty }H^{0}(X,L_{m\chi}^{\otimes cm})^{\hat{U}} \textit{ and } \oplus _{m=0}^{\infty }H^{0}(X,L_{m\chi}^{\otimes cm})^{H} = (\oplus _{m=0}^{\infty }H^{0}(X,L_{m\chi}^{\otimes cm})^{\hat{U}})^{R} $$are finitely generated;
-
2.
the enveloping quotient \(X\! \mathbin {\textit{\rotatebox [origin=c]{70}{\scalebox {1.2}{$\approx $}}}} \!\hat{U}\) is the projective variety associated to the algebra of invariants \(\oplus _{m=0}^{\infty }H^{0}(X,L_{m\chi}^{\otimes cm})^{\hat{U}}\) and is a geometric quotient of the open subset \(X^{s,\hat{U}}_{\min +}\) of \(X\) by \(\hat{U}\);
-
3.
the enveloping quotient \(X\! \mathbin {\textit{\rotatebox [origin=c]{70}{\scalebox {1.2}{$\approx $}}}} \!H\) is the projective variety associated to the algebra of invariants \(\oplus _{m=0}^{\infty }H^{0}(X,L_{m\chi}^{\otimes cm})^{{H}}\) and is the classical GIT quotient of \(X \! \mathbin {\textit{\rotatebox [origin=c]{70}{\scalebox {1.2}{$\approx $}}}} \!\hat{U}\) by the induced action of \(R/\lambda ({\mathbb{G} }_{m})\) with respect to the linearisation induced by a sufficiently divisible tensor power of \(L\).
Remark 3.7
The reason that it is only a sufficiently divisible tensor power of \(L\) which induces a line bundle on the geometric quotient \(X^{s,\hat{U}}_{\min +}/\hat{U}\) is that the action of \(\hat{U}\) on \(X^{s,\hat{U}}_{\min +}\) may have nontrivial (but finite) stabilisers. Replacing \(L\) by \(L^{\otimes N}\) where \(N\) is sufficiently divisible ensures that \(\operatorname{Stab}_{\hat{U}}(x)\) acts trivially on the fibre of \(L\) over \(x\) for each \(x \in X^{s,\hat{U}}_{\min +}\).
Definition 3.8
When the conditions of Theorem 3.6 hold, we call \(X \! \mathbin {\text{\rotatebox [origin=c]{70}{\scalebox {1.2}{$\approx $}}}} \!H\) (respectively \(X \! \mathbin {\text{\rotatebox [origin=c]{70}{\scalebox {1.2}{$\approx $}}}} \!\hat{U}\)) a GIT quotient and we denote it by \(X/\!/H\) (respectively \(X /\!/ \hat{U}\)).
Definition 3.9
Let \((X,L,H,\hat{U},\chi ,c)\) be ingredients for a well-adapted linear action (as in Definition 3.5) such that (∗) holds (as in Definition 3.4). We denote by \(X^{s,{{H}}}_{\min +}\) and \(X^{ss,{{H}}}_{\min +}\) the pre-images in \(X^{s,{{\hat{U}}}}_{\min +} \) of the stable and semistable loci for the induced linear action of the reductive group \(H/\hat{U}= R/\lambda ({\mathbb{G} }_{m})\) on \(X/\!/ \hat{U} = X^{s,\hat{U}}_{\min +}/\hat{U}\). We denote by \(Z_{\min}(X)^{s(s),R/\lambda ({\mathbb{G} }_{m})}\) the (semi)stable locus for the action of the reductive group \(R/\lambda ({\mathbb{G} }_{m})\) on \(Z_{\min}(X)\), with the (rational) linearisation induced by twisting the original linearisation of the \(R\)-action by a rational multiple of \(\chi \) chosen so that after twisting \(\lambda ({\mathbb{G} }_{m})\) acts trivially on the restriction of \(L\) to \(Z_{\min}(X)\) (equivalently the twisted linearisation is borderline adapted in the sense of Definition 3.2). Then we say that
-
(i)
weak \(H\)-stability\({=}H\)-semistability holds if \(X^{s,{{H}}}_{\min +} = X^{ss,{{H}}}_{\min +}\), and
-
(ii)
strong \(H\)-stability\({=}H\)-semistability holds if
$$ Z_{\min}(X)^{s,R/\lambda ({\mathbb{G} }_{m})} = Z_{\min}(X)^{ss,R/ \lambda ({\mathbb{G} }_{m})} \neq \emptyset . $$
Remark 3.10
These versions both coincide with condition (∗) when \(H=\hat{U}\), which will be the case in the situation considered in this paper. In general the weak version can depend on the choice of the character \(\chi \), whereas the strong version is independent of this choice since \(\lambda ({\mathbb{G} }_{m})\) acts trivially on \(Z_{\min}(X)\) and on the restriction of \(L\) to \(Z_{\min}(X)\).
Theorem 3.11
[10] Thm 4.28
Let \((X,L,H,\hat{U},\chi ,c)\) be ingredients for a well-adapted linear action (as in Definition 3.5) such that semistability coincides with stability for the action of \(H\) in the strong sense of Definition 3.9. Suppose also that \(UZ_{\min}(X)\) is not dense in \(X\). Then the projective variety \(X/\!/ H\) is a geometric quotient of \(X^{s,{{H}}}_{\min +}\) by the action of \(H\), the stabiliser \(\operatorname{Stab}_{H}(x)\) is finite for all \(x \in X^{s,{{H}}}_{\min +}\) and
where \(p:X^{0}_{\min} \to Z_{\min}(X)\) is as in (3.1) and \(Z_{\min}(X)^{s,R/\lambda ({\mathbb{G} }_{m})}\) is as in Definition 3.9.
Remark 3.12
Here by Theorem 3.6\(X/\!/ H\) is the projective variety associated to the finitely generated graded algebra \(\oplus _{m=0}^{\infty }H^{0}(X,L_{m\chi}^{\otimes cm})^{H} \). Thus \(X/\!/ H\) has an ample line bundle which we will denote by \(\mathcal{O}_{X/\!/ H}(1)\) whose pullback to \(p^{-1}(Z_{\min}(X)^{s,R/\lambda ({\mathbb{G} }_{m})}) \setminus UZ_{ \min}(X)\) is isomorphic to \(L^{\otimes N}\) for some \(N \geq 1\), and such that when \(N\) divides \(cm\) then \(H\)-invariant sections of \(L_{m\chi}^{\otimes cm}\) define sections of \(\mathcal{O}_{X/\!/H}(cm/N)\). More precisely, sections of \(\mathcal{O}_{X/\!/H}(cm/N)\) pull back to \(H\)-invariant sections of the restriction of \(L_{m\chi}^{\otimes cm}\) to \(X^{s,H}_{\min +}\), and these all extend to \(H\)-invariant sections of \(L_{m\chi}^{\otimes cm}\) over \(X\); moreover every \(H\)-invariant section of \(L_{m\chi}^{\otimes cm}\) over \(X\) vanishes on the complement of \(X^{s,H}_{\min +}\).
Note that although \(H\) acts on \(p^{-1}(Z_{\min}(X)^{s,R/\lambda ({\mathbb{G} }_{m})}) \setminus UZ_{ \min}(X)\) with finite stabilisers it may not act freely, and if \(x \in p^{-1}(Z_{\min}(X)^{s,R/\lambda ({\mathbb{G} }_{m})}) \setminus UZ_{\min}(X)\) then \(\mathrm{Stab}_{H}(x)\) may not act trivially on the fibre \(L_{x}\) of the line bundle \(L\). However if \(N\) is chosen to be sufficiently divisible then \(\mathrm{Stab}_{H}(x)\) will act trivially on \(L_{x}^{\otimes N}\) for all such \(x\), so that \(L^{\otimes N}\) descends to a line bundle on \(X/\!/H\).
Remark 3.13
If \(UZ_{\min}(X)\) is dense in \(X\), then \(Z_{\min}(X)\) is the natural candidate for the role of quotient of \(X\) by \(U\) or \(\hat{U}\), with semistable locus \(X^{0}_{\min}\), via the map \(p:X^{0}_{\min} \to Z_{\min}(X)\) defined at (3.1), and \(Z_{\min}(X)/\!/(R/\lambda ({\mathbb{G} }_{m}))\) is the natural candidate for the quotient of \(X\) by \(H\). Thus in this case we are reduced to reductive GIT.
In the set-up of the Green–Griffiths–Lang conjecture, the group which acts is a polynomial reparametrisation group of the form \(\hat{U}=U \rtimes {\mathbb{C} }^{*}\), but initially the condition (∗) that semistability coincides with stability in Definition 3.4 is not satisfied. However we can blow up to obtain a situation in which semistability does coincide with stability and then apply Theorem 3.6 to construct a non-reductive quotient; we can also apply Theorems 4.4-4.7 below in order to integrate over this non-reductive quotient. In fact this blow-up construction works much more generally; see [15].
4 Moment maps and cohomology of non-reductive quotients
In this section we briefly summarise the results of [10], which generalise results of Martin [32] to the cohomology of GIT quotients by non-reductive groups with internally graded unipotent radicals.
4.1 Reductive actions
First let us recall the reductive picture. Let \(X\) be a nonsingular complex projective variety acted on by a complex reductive group \(G\) with respect to an ample linearisation. Then we can choose a maximal compact subgroup \(K\) of \(G\) and a \(K\)-invariant Fubini–Study Kähler metric on \(X\) with corresponding moment map \(\mu :X\to \mathfrak{k}^{*}\), where \(\mathfrak{k}\) is the Lie algebra of \(K\) and \(\mathfrak{k}^{*} = \,{\mathrm{Hom}} \,_{{\mathbb{R} }}(\mathfrak{k},{ \mathbb{R} })\) is its dual. \(\mathfrak{k}^{*}\) embeds naturally in the complex dual \(\mathfrak{g}^{*} = \,{\mathrm{Hom}} \,_{{\mathbb{C} }}(\mathfrak{k},{ \mathbb{C} })\) of the Lie algebra \(\mathfrak{g} = \mathfrak{k} \otimes {\mathbb{C} }\) of \(G\), as \(\mathfrak{k}^{*} = \{\xi \in \mathfrak{g}^{*}: \xi (\mathfrak{k}) \subseteq {\mathbb{R} }\}\); using this identification we can regard \(\mu : X \to \mathfrak{g}^{*}\) as a ‘moment map’ for the action of \(G\), although of course it is not a moment map for \(G\) in the traditional sense of symplectic geometry.
In [29] it is shown that the norm-square \(f=||\mu ||^{2}\) of the moment map \(\mu :X\to \mathfrak{k}^{*}\subseteq \mathfrak{g}^{*}\) induces an equivariantly perfect Morse stratification of \(X\) such that the open stratum which retracts equivariantly onto the zero level set \(\mu ^{-1}(0)\) of the moment map coincides with the GIT semistable locus \(X^{ss}\) for the linear action of \(G\) on \(X\). In particular this tells us that the restriction map
is surjective; we also have an isomorphism (of vector spaces though not of algebras) \(H^{*}_{G}(X;{\mathbb{Q} }) \cong H^{*}(X;{\mathbb{Q} }) \otimes H^{*}(BG;{ \mathbb{Q} })\). Moreover, \(\mu ^{-1}(0)\) is \(K\)-invariant and its inclusion in \(X^{ss}\) induces a homeomorphism
When \(X^{s}=X^{ss}\) the \(G\)-equivariant rational cohomology of \(X^{ss}\) coincides with the ordinary rational cohomology of its geometric quotient \(X^{ss}/G\), which is the GIT quotient \(X/\!/G\), and we get expressions for the Betti numbers of \(X/\!/G\) in terms of the equivariant Betti numbers of \(X\) and the equivariant Betti numbers of the unstable GIT strata, which can be described inductively [29]. In order to obtain the ring structure on the rational cohomology of \(X/\!/G\), the surjectivity of the composition
can be combined with Poincaré duality on \(X/\!/G\) and the nonabelian localisation formulas for intersection pairings on \(X/\!/G\) given in [28].
Martin [32] used (4.1) to obtain formulas for the intersection pairings on the quotient \(X/\!/G\) in a different way, by relating these pairings to intersection pairings on the associated quotient \(X/\!/T_{{\mathbb{C} }}\), where \(T_{{\mathbb{C} }}\) is a maximal torus for \(G\). He proved a formula expressing the rational cohomology ring of \(X/\!/G\) in terms of the rational cohomology ring of \(X/\!/T_{{\mathbb{C} }}\) and an integration formula relating intersection pairings on the cohomology of \(X/\!/G\) to corresponding pairings on \(X/\!/T_{{\mathbb{C} }}\). This integration formula, combined with methods from abelian localisation, leads to residue formulas for pairings on \(X/\!/G\) which are closely related to those of [28] (see also [44]).
Note that, in the symplectic approach leading to (4.1), a maximal compact subgroup \(K\) of \(G\) and a \(K\)-invariant Fubini–Study Kähler metric \(\omega \) are fixed and the corresponding moment map \(\mu :X \to \mathfrak{k}^{*}\) for the \(K\)-action is used. In [10] we introduced a new point of view as follows. Any other maximal compact subgroup of \(G\) is given by \(K' = g^{-1}Kg\) for some \(g \in G\); then \(\omega _{K'} = g^{*}\omega \) is \(K'\)-invariant and if \(\mu _{K}:X \to \mathfrak{k}^{*}\) exists then \(\mu _{K'} = Ad^{*}(g^{-1}) \circ \mu _{K} \circ g\) is a \(K'\)-moment map with respect to \(\omega _{K'}\), where \(Ad^{*}\) denotes the co-adjoint action of \(G\) on the complex dual \(\mathfrak{g}^{*}\) of its Lie algebra and \(\mathfrak{k}^{*}\) is embedded in \(\mathfrak{g}^{*}\) as above. So to define a ‘moment map’ for the \(G\)-action on \(X\), instead of fixing a Fubini-Study Kähler form \(\omega \) it is natural to ask for a \(G\)-orbit \(\Omega \) in
where \(\text{K\"{a}hler}(X)\) is the space of Kähler forms on the complex manifold \(X\) and
In [10] we call such a \(G\)-orbit \(\Omega =\{(g^{-1}Kg,g^{*} \omega ): g\in G\}\) a \(G\)-equivariant Kähler structure on \(X\). We define an \(\Omega \)-moment map for the \(G\)-action on \(X\) to be a smooth \(G\)-equivariant map
such that \(\mathfrak{m}_{G,X,\Omega }(K,\omega , x) = \mu _{(K,\omega )}(x)\) for each \((K,\omega ) \in \Omega \) and \(x \in X\), where \(\mu _{(K,\omega )}:X \to \mathfrak{g}^{*}\) is the composition of a moment map for the \(K\)-action on \((X,\omega )\) with the canonical embedding
of the dual of the Lie algebra of \(K\) in \(\mathfrak{g}^{*}\). Given a \(G\)-equivariant Kähler structure \(\Omega \) on \(X\) and a choice of maximal compact subgroup \(K\) of \(G\) with \((K,\omega ) \in \Omega \), the existence and choice of an \(\Omega \)-moment map \(\mathfrak{m}_{G,X,\Omega }: \Omega \times Y \to \mathfrak{g}^{*}\) for the \(G\)-action on \(X\) is equivalent to the existence and choice of a moment map \(\mu :X \to {{\mathfrak {k}}}^{*}\) in the traditional sense for the \(K\)-action on the Kähler manifold \((X,\omega )\). However, this point of view with emphasis on the complex group action rather than the compact one, is a more natural one to take when extending the notion of a moment map to non-reductive linear algebraic groups.
4.2 Non-reductive actions
In [10] similar results are obtained for non-reductive actions. Let \(X \subset {\mathbb{P} }^{n}\) be a nonsingular complex projective variety with a linear action (with respect to an ample line bundle \(L\)) of a complex linear algebraic group \(H=U \rtimes R\) with internally graded unipotent radical \(U\). Let \(\hat{U}=U \rtimes \lambda ({\mathbb{C} }^{*}) \subseteq H\) where \(\lambda : {\mathbb{C} }^{*} \to Z(R)\) is a grading 1-parameter subgroup; then \(\lambda (S^{1}) \subseteq \lambda ({\mathbb{C} }^{*}) \subseteq \hat{U}\) is a maximal compact subgroup of \(\hat{U}\). Suppose that \(H\) acts on \(X\) via a homomorphism \(\rho : H \rightarrow G = \mathrm{GL}(n+1, {\mathbb{C} })\), and the Levi subgroup \(R\) is the complexification of a maximal compact subgroup \(Q=K\cap H\) of \(H\), where the unitary group \(K=U(n+1)\) is a maximal compact subgroup of \(G\). So \(Q\) preserves the standard \(U(n+1)\)-invariant hermitian inner product \(\langle \, ,\rangle \) on \({\mathbb{C} }^{n+1}\). We can use the corresponding Fubini-Study form \(\omega \) on \({\mathbb{P} }^{n}\) to define a \(G\)-equivariant Kähler structure
on \({\mathbb{P} }^{n}\) which contains \((K,\omega )\). We can restrict the corresponding \(\Omega \)-moment map \(\mathfrak{m}_{G,X,\Omega }: \Omega _{G, {\mathbb{P} }^{n}} \times { \mathbb{P} }^{n} \to {\mathfrak {g}}^{*}\) to \(\Omega _{H,X} \times X\) where
and compose it with the projection \(p^{*}: {\mathfrak {g}}^{*} \to \mathfrak{h}^{*}\) to obtain an \(\Omega \)-moment map \(\Omega _{H,X} \times X \to \mathfrak{h}^{*}\) for the \(H\)-action on \(X\) fitting into the following diagram:
We obtain another \(\Omega \)-moment map by adding to \(\mathfrak{m}_{H,X,\Omega }\) any \(H\)-equivariant map \(\Omega \times X \to \mathfrak{h}^{*}\) which is constant and \(K\cap H\)-invariant on \(\{(K,\omega )\} \times X\) for every \((K,\omega )\) in the \(H\)-orbit \(\Omega \). For \((K,\omega ) \in \Omega _{H,X}\) the restriction \(\mu ^{H}_{(K,\omega )}: X \to \mathfrak{h}^{*}\) to \(\{(K,\omega )\} \times X\) is given, up to the addition of a \(K \cap H\)-invariant constant, by
where ⋅ denotes the natural pairing between \(\mathfrak{h}^{*}\) and \(\mathfrak{h}\).
Suppose now that the linearisation of the action of \(H\) on \(X\) is well-adapted and \(H\)-stability\({=}H\)-semistability in the strong sense of Definition 3.9). Then it is shown in [10] that for any \((K,\omega ) \in \Omega _{X,H}\)
-
1.
\(X^{s,H}_{\min +} = X^{ss,H}_{\min +} = \{ x \in X:0 \in \mathfrak{m}_{H,X, \Omega}(\Omega \times \{x\})\}=H(\mu _{(K,\omega )}^{H})^{-1}(0)\),
-
2.
0 is a regular value of \(\mu _{(K,\omega )}^{H}\) and
-
3.
the embedding of \((\mu _{(K,\omega )}^{H})^{-1}(0) \) in \(X^{s,H}_{\min +}\) induces a diffeomorphism of orbifolds
$$ (\mu _{(K,\omega )}^{H})^{-1}(0)/(K \cap H) \to X^{s,H}_{\min +}/H = X/ \!/H. $$(4.4)
Remark 4.1
In [10] similar results are obtained in the more general situation when \(X\) is compact Kähler but not necessarily projective.
In the present paper we will work with actions of the diffeomorphism group (see §2.3)
on projective varieties. Therefore we only state the remaining results of [10] for the situation when \(H=\hat{U}=U \rtimes \lambda ({\mathbb{C} }^{*})\) acts via a homomorphism \(\rho : \hat{U}\to G = \mathrm{GL}(n+1)\) on a nonsingular projective variety \(X \subset {\mathbb{P} }^{n}\), such that if \(K=U(n+1)\) then \(K \cap \hat{U}=\lambda (S^{1})\) is the unique maximal compact subgroup of the one-parameter subgroup \(\lambda ({\mathbb{C} }^{*})\). The Lie algebra of \(\hat{U}\) decomposes as a real vector space as
where \(\mathrm{Lie}(K \cap \hat{U})=i {\mathbb{R} }\) and \({\mathfrak {u}}\) is the Lie algebra of the complex unipotent group \(U\). Diagram (4.3) becomes
We can allow the addition of a rational character (that is, a rational multiple of the derivative of the group homomorphism \(\hat{U}\to {\mathbb{C} }^{*}\) with kernel \(U\)) to this ‘\(\Omega \)-moment map’ \(\mathfrak{m}_{\hat{U},X,\Omega }:X \to \hat{{\mathfrak {u}}}^{*}\). Indeed, in the Kähler setting there is no reason not to allow real multiples, rather than only rational ones, here.
In this case, if \(\hat{U}\)-stability\({=}\hat{U}\)-semistability holds in the sense of Definition 3.4 then (4.4) tells that for any \((K,\omega ) \in \Omega _{\hat{U},X}\) the embedding \((\mu _{(K,\omega )}^{\hat{U}})^{-1}(0) \hookrightarrow X^{s,\hat{U}}_{ \min +}\) induces a diffeomorphism of orbifolds
where \(S^{1}=\lambda (S^{1})=K \cap \hat{U}\). We fix \((K,\omega )\in \Omega _{\hat{U},X}\), and drop it from the notation in the rest of this section, writing \(\mu _{\hat{U}}\) for \(\mu _{(K,\omega )}^{\hat{U}}\) and \(\mu _{\lambda ({\mathbb{C} }^{*})}\) fpr \(\mu _{(K,\omega )}^{\lambda ({\mathbb{C} }^{*})}\) (which is a moment map in the usual sense for \(\lambda (S^{1})\) acting on \(X\).
Following Martin (see diagram (1.2) in [10]) we consider
Definition 4.2
For a weight \(\alpha \) of \(\lambda ({\mathbb{C} }^{*}) \subseteq \hat{U}\), let \({\mathbb{C} }_{\alpha }\) denote the corresponding 1-dimensional complex representation of \({\mathbb{C} }^{*}\) and let
denote the associated line bundle whose Euler class is denoted by \(e(\alpha ) \in H^{2}(X/\!/ {\mathbb{C} }^{*})\simeq H_{{\mathbb{C} }}^{2}(X)\). For a \({\mathbb{C} }^{*}\)-invariant complex subspace \(\mathfrak{a} \subseteq {\mathfrak {u}}\) let
denote the corresponding vector bundle.
Proposition 4.3
[10], Proposition 7.12 (i)
The vector bundle \(V_{{\mathfrak {u}}}^{*} \to X/\!/{\mathbb{C} }^{*}\) has a \(C^{\infty}\)-section \(s\) which is transverse to the zero section and whose zero set is the submanifold \(\mu _{\hat{U}}^{-1}(0)/S^{1} \subseteq X/\!/{\mathbb{C} }^{*}\). Therefore the \({\mathbb{C} }^{*}\)-equivariant normal bundle is
This leads in [10] to the following theorems:
Theorem 4.4
Let \(X\) be a smooth projective variety endowed with a well-adapted action of \(\hat{U}=U \rtimes {\mathbb{C} }^{*}\) such that \(\hat{U}\)-stability\({=} \hat{U}\)-semistability holds. Then there is a natural ring isomorphism
Here \(\mathrm{Euler}(V^{*}_{{\mathfrak {u}}}) \in H^{*}(X/\!/{\mathbb{C} }^{*})\) is the Euler class of the bundle \(V^{*}_{{\mathfrak {u}}}\) and
is the annihilator ideal.
Theorem 4.5
[10], Theorem 1.5
Let \(X\) be a smooth projective variety endowed with a well-adapted action of \(\hat{U}=U \rtimes {\mathbb{C} }^{*}\) such that \(\hat{U}\)-stability\({=} \hat{U}\)-semistability holds. Assume for simplicity that the stabiliser in \(\hat{U}\) of a generic \(x \in X\) is trivial. Given a cohomology class \(a \in H^{*}(X/\!/\hat{U})\) with a lift \(\tilde{a}\in H^{*}(X/\!/{\mathbb{C} }^{*})\), then
where \(\mathrm{Euler}(V^{*}_{{\mathfrak {u}}})\) is the cohomology class defined in Theorem 4.4. Here we say that \(\tilde{a}\in H^{*}(X/\!/{\mathbb{C} }^{*})\) is a lift of \(a\in H^{*}(X/\!/\hat{U})\) if \(a=i^{*}\tilde{a}\).
Remark 4.6
Theorem 4.5 can be generalised to allow the triviality assumption for the stabiliser in \(\hat{U}\) of a generic \(x \in X\) to be omitted; then the formula for \(\int _{X/\!/\hat{U}}a\) is multiplied by a strictly positive rational depending on the sizes of the stabilisers in \(\hat{U}\) and \({\mathbb{C} }^{*}\) of a generic \(x \in X\).
Finally, we have residue formulas for the intersection pairings on the quotient \(X/\!/{\mathbb{C} }^{*}\). Note that \(\hat{U}\) is homotopy equivalent to \({\mathbb{C} }^{*}\) and to \(S^{1}\) so that \(\hat{U}\)-equivariant cohomology is isomorphic to \({\mathbb{C} }^{*}\)-equivariant cohomology and to \(S^{1}\)-equivariant cohomology. There are two surjective ring homomorphisms
from the \(S^{1}\)-equivariant cohomology of \(X\) to the ordinary cohomology of the corresponding GIT quotients. The fixed points of the maximal compact subgroup \(S^{1}\) of \(\hat{U}\) on \(X\subseteq {\mathbb{P} }^{n}\) correspond to the weights of the \({\mathbb{C} }^{*}\) action on \(X\), and since this action is well-adapted, these weights satisfy
We can represent elements of \(H_{\hat{U}}^{*}(X;{\mathbb{Q} })=H_{S^{1}}^{*}(X;{\mathbb{Q} })\) as polynomial functions on the Lie algebra of \({\mathbb{C} }^{*}\) whose coefficients are differential forms on \(X\) and which are equivariantly closed.
Theorem 4.7
[10], Theorem 7.15 and Corollary 7.16
Let \(X\) be a nonsingular projective variety (assumed connected) endowed with a well-adapted action of \(\hat{U}=U \rtimes {\mathbb{C} }^{*}\) such that \(\hat{U}\)-stability\({=}\hat{U}\)-semistability holds (in the sense of Definitions 3.4and 3.5). Let \(z\) be the standard coordinate on the Lie algebra of \({\mathbb{C} }^{*}\). Given any \(\hat{U}\)-equivariant cohomology class \(\eta \) on \(X\) represented by an equivariant differential form \(\eta (z)\) whose degree is the dimension of \(X/\!/\hat{U}\), we have
where \(F_{\min} = Z_{\min}(X)\) is the union of those connected components of the fixed point locus \(X^{{\mathbb{C} }^{*}}\) on which the \(S^{1}\)-moment map takes its minimum value \(\omega _{\min}\), and \(n_{\hat{U}}\) is a strictly positive rational number which depends only on \(\hat{U}\) and the size of the stabiliser in \(\hat{U}\) of a generic \(x \in X\). Here \(\mathcal{N}_{F_{\min}}\) is the normal bundle to \(F_{\min}\) in \(X\) and \(V^{*}_{{\mathfrak {u}}}\to F_{\min}\) is the equivariant vector bundle given by \(F_{\min} \times {\mathfrak {u}}\).
Remark 4.8
In the situation of Theorem 4.7, we can rewrite this formula as
where \(\mathcal{N}_{UF_{\min}}\) is the normal to \(UF_{\min}\) in \(X\), since the restriction to \(F_{\min}\) of the tangent bundle to \(UF_{\min}\) is the direct sum of the tangent bundle to \(F_{\min}\) and the trivial bundle \(V_{{\mathfrak {u}}}\) with fibre the Lie algebra of \(U\). We can also identify \(F_{\min}\) with \(UF_{\min}/U\) and identify \(\mathcal{N}_{UF_{\min}}\) with the normal to \(UF_{\min}/U\) in \(X^{0}_{\min}/U\).
Notice in addition that the surjection \(\kappa _{\hat{U}}: H_{\hat{U}}^{*}(X;{\mathbb{Q} })=H_{S^{1}}^{*}(X;{ \mathbb{Q} }) \to H^{*}(X/\!/\hat{U}; {\mathbb{Q} })\) is the composition of two surjections
and that \(F_{\min} \subseteq X^{0}_{\min}\). Thus to calculate any intersection pairing
using (4.8), we only need to know the restriction to \(X^{0}_{\min}\) of \(S^{1}\)-equivariant cohomology classes \(\alpha \) and \(\beta \) which represent \(\kappa _{\hat{U}}(\alpha )\) and \(\kappa _{\hat{U}}(\beta )\).
5 Non-reductive GIT compactification of the jet differentials bundle
Let \(X\) be a nonsingular complex projective variety of dimension \(n\). In this section we will use non-reductive GIT to construct a projective completion of the quasi-projective quotient \(J_{k}^{\mathrm{reg}}X/\mathrm{Diff}_{k}(1)\), introduced in §2.3. Recall that \(\mathrm{Diff}_{X}\) denotes the principal \(\mathrm{Diff}_{k}(n)\)-bundle over \(X\) formed by all local polynomial coordinate systems on \(X\), and that
where \(X_{k}^{\text{reg}}=J_{k}^{\text{reg}}(1,n)/\mathrm{Diff}_{k}(1))\) is isomorphic to each fibre of \(\mathcal{X}_{k}^{\text{reg}}\) over \(X\). We will construct a projective completion
of \(\mathcal{X}_{k}^{\text{reg}}\) where the fibre \(X_{k}^{\mathrm{GIT}}\) of \(\mathcal{X}_{k}^{\mathrm{GIT}}\) over \(X\) is a non-reductive GIT quotient of a blow-up of the projective space
by the action of
Here \(v_{1},\ldots , v_{k} \in {\mathbb{C} }^{n}\) are vectors representing the columns of a matrix \(M \in \,{\mathrm{Hom}} \,({\mathbb{C} }^{k}, {\mathbb{C} }^{n})\), and \(x\) is the compactifying coordinate, while \(\hat{U}\) acts via the right action
or equivalently via the left action
Remark 5.1
We need to be careful here with the construction of \(\mathcal{X}_{k}^{\mathrm{GIT}}\) as \(\mathrm{Diff}_{X} \times _{\mathrm{Diff}_{k}(n)} X_{k}^{ \mathrm{GIT}}\). This is because \(\mathrm{Diff}_{k}(n)\) does not act naturally on the projective space \({\mathbb{P} }({\mathbb{C} }\oplus J_{k}(1,n)) = {\mathbb{P} }({ \mathbb{C} }\oplus \,{\mathrm{Hom}} \,({\mathbb{C} }^{k}, {\mathbb{C} }^{n}))\) or its blow-up, since its action on \(J_{k}(1,n)\) is not linear, and the identification of a fibre of \(J_{k}X\) with \({\mathrm{Hom}} \,({\mathbb{C} }^{k}, {\mathbb{C} }^{n}))\) depends on a choice of local coordinates on \(X\). Nonetheless we will find that \(\mathrm{Diff}_{k}(n)\) does act on \(X_{k}^{\mathrm{GIT}}\) compatibly with its action on \(J_{k}^{\mathrm{reg}}(1,n)/\mathrm{Diff}_{k}(1)\).
We want to apply the results of non-reductive GIT described in \(\S \)3, which are stated for left actions. For this we need a one-parameter subgroup \(\lambda :{\mathbb{C} }^{*} \to \hat{U}\) whose adjoint action on the Lie algebra of \(U\) has only strictly positive weights; we can take
and then
The weights of this (left) action of the one-parameter subgroup of \(\hat{U}\) defined by \(\lambda \) are \(\{0, 1, 2, \ldots , k\}\). The minimal weight space is the point
and the \(U\)-stabiliser of this point is \(U\), so this action does not satisfy the ‘semistability coincides with stability’ condition (see Definition 3.4) which is required for the main results described in \(\S \S \)3 and 4. We wish to construct a blow-up of the projective space \({\mathbb{P} }({\mathbb{C} }\oplus J_{k}(1,n)) = {\mathbb{P} }({ \mathbb{C} }\oplus \,{\mathrm{Hom}} \,({\mathbb{C} }^{k}, {\mathbb{C} }^{n}))\) in order to satisfy this condition. Following [15] we try blowing up the projective space \({\mathbb{P} }({\mathbb{C} }\oplus \,{\mathrm{Hom}} \,({\mathbb{C} }^{k}, { \mathbb{C} }^{n}))\) along \(Z_{\min}\) to get
embedded in \({\mathbb{P} }^{kn} \times {\mathbb{P} }^{kn-1} \subseteq {\mathbb{P} }^{(kn+1)kn-1}\). We fix an ample linearisation \(L=\mathcal{O}_{ {\mathbb{P} }^{kn}}(1) \otimes \mathcal{O}_{ { \mathbb{P} }^{kn-1}}(1)\) on \({\mathbb{P} }^{kn} \times {\mathbb{P} }^{kn-1}\) and restrict it to \({\tilde{\mathbb{P} }}\). The minimal weight space for the action of \(\lambda ({\mathbb{C} }^{*})\) on \({\tilde{\mathbb{P} }}\) is the intersection \(\tilde{Z}_{\min }\) of the exceptional divisor \(E\) and the strict transform of \({\mathbb{P} }[x:v_{1}:0:\cdots :0] \subseteq {\mathbb{P} }({\mathbb{C} } \oplus \,{\mathrm{Hom}} \,({\mathbb{C} }^{k}, {\mathbb{C} }^{n}))\):
The \(U\)-stabiliser of any point in \(\tilde{Z}_{\min }\) is trivial, and hence stability coincides with semistability for the induced \(\hat{U}\) action on \({\tilde{\mathbb{P} }}\). Thus we can apply Theorem 3.6 to obtain a non-reductive GIT quotient
with respect to a well-adapted shift of \(L\). Then \({\tilde{\mathbb{P} }}/\!/\hat{U}\) is a projective variety and is a geometric quotient by \(\hat{U}\) of the open subvariety \({\tilde{\mathbb{P} }}^{s,\hat{U}} = {\tilde{\mathbb{P} }}^{0}_{\min} \setminus U Z_{\min}( {\tilde{\mathbb{P} }})\) of \({\tilde{\mathbb{P} }}\), which contains \(J_{k}^{\text{reg}}(1,n)\). Thus \({\tilde{\mathbb{P} }}/\!/\hat{U}\) is a projective completion of \(J_{k}^{\text{reg}}(1,n)/\mathrm{Diff}_{k}(1)\).
Remark 5.2
The open subvariety \({\tilde{\mathbb{P} }}^{s,\hat{U}} = {\tilde{\mathbb{P} }}^{0}_{\min} \setminus U Z_{\min}( {\tilde{\mathbb{P} }})\) of \({\tilde{\mathbb{P} }}\) contains \(J_{k}^{\text{reg}}(1,n)\) and is contained in the blow-up of the affine space \(J_{k}(1,n)\cong \,{\mathrm{Hom}} \,({\mathbb{C} }^{k}, {\mathbb{C} }^{n})\) at the origin; here the origin represents the trivial jet which is identically 0. The action of \(\mathrm{Diff}_{k}(n)\) on \(J_{k}(1,n)\) is not linear but it fixes the origin and hence there is an induced action on the blow-up of \(J_{k}(1,n)\) at the origin. This action preserves \({\tilde{\mathbb{P} }}^{s,\hat{U}} = {\tilde{\mathbb{P} }}^{0}_{\min} \setminus U Z_{\min}( {\tilde{\mathbb{P} }})\) and commutes with the action of \(\mathrm{Diff}_{k}(1)\), so there is an induced action of \(\mathrm{Diff}_{k}(n)\) on the geometric quotient \(X_{k}^{\mathrm{GIT}}= {\tilde{\mathbb{P} }}/\!/\hat{U}= { \tilde{\mathbb{P} }}^{s,\hat{U}}/\hat{U}\).
Definition 5.3
Let \(\mathcal{X}_{k}^{\mathrm{GIT}} = \mathrm{Diff}_{X} \times _{ \mathrm{Diff}_{k}(n)} X_{k}^{\mathrm{GIT}} \) where \(X_{k}^{\mathrm{GIT}}\) is defined at (5.1) above.
Thus \(\mathcal{X}_{k}^{\mathrm{GIT}}\) fibres over \(X\) with fibre \(X_{k}^{\mathrm{GIT}} \), and can be regarded as a projective completion of \(\mathcal{X}_{k}^{\text{reg}}=J_{k}^{\text{reg}}X/\mathrm{Diff}_{k}(1) \cong \mathrm{Diff}_{X} \times _{\mathrm{Diff}_{k}(n)} X_{k}^{ \text{reg}}\). It can also be identified with the geometric quotient by \(\mathrm{Diff}_{k}(1)\) of an open subvariety of the blow-up of the jet bundle \(J_{k}X\) along its zero section. The jet bundle \(J_{k}X\) is not a vector bundle over \(X\), since its structure group is \(\mathrm{Diff}_{k}(n)\) which is not a subgroup of \(\mathrm{GL}(n)\), but nevertheless it has a well-defined zero section.
Remark 5.4
Recall from Remark 3.12 that we have fixed a positive integer \(N\) such that there is an ample line bundle \(\mathcal{O}_{X_{k}^{\mathrm{GIT}}}(1)\) on \(X_{k}^{\mathrm{GIT}}\) which pulls back to \(L^{\otimes N}\) on \({\tilde{\mathbb{P} }}^{s,\hat{U}}\). Here \(\mathcal{O}_{X_{k}^{\mathrm{GIT}}}(1)\) can be constructed as the geometric quotient of the restriction of \(L^{\otimes N}\) to \({\tilde{\mathbb{P} }}^{s,\hat{U}}\) by the well-adapted linear action of \(\mathrm{Diff}_{k}(1)\), where \(N\) is sufficiently divisible that for every \(x \in {\tilde{\mathbb{P} }}^{s,\hat{U}}\) the finite stabiliser \(\operatorname{Stab}_{\hat{U}}(x)\) acts trivially on the fibre of \(L^{\otimes N}\) at \(x\). Then sections of powers of \(L^{\otimes N}\) which are \(\hat{U}\)-invariant for the well-adapted linearisation define sections of powers of \(\mathcal{O}_{X_{k}^{\mathrm{GIT}}}(1)\). More precisely, if \(j \) is a natural number, sections of \(\mathcal{O}_{X_{k}^{\mathrm{GIT}}}(j)\) pull back to \(\hat{U}\)-invariant sections of the restriction of \(L^{\otimes Nj}\) to \({\tilde{\mathbb{P} }}^{s,\hat{U}}\), and these all extend to \(\hat{U}\)-invariant sections of \(L^{\otimes Nj}\) over \({\tilde{\mathbb{P} }}\); here every \(\hat{U}\)-invariant section of \(L^{\otimes Nj}\) over \({\tilde{\mathbb{P} }}\) vanishes on the complement of \({\tilde{\mathbb{P} }}^{s,\hat{U}}\). Since \({\tilde{\mathbb{P} }}^{s,\hat{U}}\) is a \(\mathrm{Diff}_{k}(n) \times \mathrm{Diff}_{k}(1)\)-invariant open subset of the blow-up of \(J_{k}(1,n)\) at 0 (where \(\hat{U}= \mathrm{Diff}_{k}(1)\)), it follows that sections of the restriction to the blow-up of \(J_{k}(1,n)\) at 0 of \(L^{\otimes Nj}\), which are \(\hat{U}\)-invariant for the well-adapted linearisation, define sections of \(\mathcal{O}_{X_{k}^{\mathrm{GIT}}}(j)\). Since \(L\) is given by \(\mathcal{O}_{ {\mathbb{P} }^{kn}}(1) \otimes \mathcal{O}_{ {\mathbb{P} }^{kn-1}}(1)\) where the restriction of the line bundle \(\mathcal{O}_{ {\mathbb{P} }^{kn}}(1)\) to \(J_{k}(1,n)\) is trivial, the restriction of \(L\) to the blow-up of \(J_{k}(1,n)\) at 0 is \(\mathcal{O}(-E)\) where \(E\) is the exceptional divisor. Furthermore sections of \(\mathcal{O}_{X_{k}^{\mathrm{GIT}}}(j)\) pull back to \(\hat{U}\)-invariant sections of the restriction of \(L^{\otimes Nj}\) to \({\tilde{\mathbb{P} }}^{s,\hat{U}}\) which extend by zero to \(\hat{U}\)-invariant sections of \(\mathcal{O}(-NjE)\) over the blow-up of \(J_{k}(1,n)\) at 0. This gives us an identification of
with a subspace of the space of \(\hat{U}\)-invariant sections of \(\bigoplus _{j=1}^{m} \mathcal{O}(-NjE)\) over the blow-up of \(J_{k}(1,n)\) at 0 which are polynomial of degree at most \(Nm\) in the coordinates on \(J_{k}(1,n)\).
Recall also that since \({\tilde{\mathbb{P} }}^{s,\hat{U}}\) is a \(\mathrm{Diff}_{k}(n) \times \mathrm{Diff}_{k}(1)\)-invariant open subset of the blow-up of \(J_{k}(1,n)\) at 0, there is an identification
where \(\tilde{J}_{k}^{s,\hat{U}}X\) is a \(\mathrm{Diff}_{k}(1)\)-invariant open subset of the blow-up \(\tilde{J}_{k}X\) of \(J_{k}X\) along its zero section. There is thus a relatively ample line bundle \(\mathcal{O}_{\mathcal{X}_{k}^{\mathrm{GIT}}}(1)\) on \(\mathcal{X}_{k}^{\mathrm{GIT}} \to X\), constructed as the geometric quotient of
by the action of \(\mathrm{Diff}_{k}(1)\), which pulls back on each fibre of \(\mathcal{X}_{k}^{\mathrm{GIT}} \to X\) to \(L^{\otimes N}\) (or equivalently to \(\mathcal{O}(-NE)\)) on \({\tilde{\mathbb{P} }}^{s,\hat{U}} = {\tilde{\mathbb{P} }}^{0}_{\min} \setminus U Z_{\min}( {\tilde{\mathbb{P} }})\). Thus sections of \(\bigoplus _{j=1}^{m} \mathcal{O}_{\mathcal{X}_{k}^{\mathrm{GIT}}}(j)\) pull back on each fibre of \(\mathcal{X}_{k}^{\mathrm{GIT}} \to X\) to restrictions of \(\hat{U}\)-invariant sections of \(\bigoplus _{j=1}^{m} \mathcal{O}(-NjE)\) over the blow-up of \(J_{k} X\) along its zero section, which are polynomial of degree at most \(Nm\) in the coordinates on the corresponding fibre of \(J_{k}X \to X\).
The first Chern class \(c_{1}(\mathcal{O}_{\mathcal{X}_{k}^{\mathrm{GIT}}}(1))\) restricts to \(c_{1}(\mathcal{O}_{X_{k}^{\mathrm{GIT}}}(1))\) on the fibre \(X_{k}^{\mathrm{GIT}}\), and this is given in the notation of Theorem 4.7 and Remark 4.8 by
where \(c_{1}^{S^{1}}(L^{\otimes N})\) is the \(S^{1}\)-equivariant first Chern class of \(L^{\otimes N}\), and \(c_{1}^{S^{1}}(L^{\otimes N}|_{ {\tilde{\mathbb{P} }}^{0}_{\min}}) = c_{1}^{S^{1}}(L^{ \otimes N})|_{ {\tilde{\mathbb{P} }}^{0}_{\min}}\) is its restriction to \({\tilde{\mathbb{P} }}^{0}_{\min}\). From Remark 5.2, since \(L=\mathcal{O}_{ {\mathbb{P} }^{kn}}(1) \otimes \mathcal{O}_{ { \mathbb{P} }^{kn-1}}(1)\) where the restriction of the line bundle \(\mathcal{O}_{ {\mathbb{P} }^{kn}}(1)\) to \(J_{k}(1,n)\) is trivial, \(c_{1}^{S^{1}}(L^{\otimes N})\) has the same restriction to the blow-up of \(J_{k}(1,n)\) as the power
of the \(S^{1}\)-equivariant first Chern class of the exceptional divisor \(E\) for the blow-up \(\tilde{J}_{k}X\) of \(J_{k}X\) along its zero section.
Now let’s consider \(X_{k}^{\mathrm{GIT}}\) in the simple cases when \(k=2,3\).
5.1 Description of \(X_{2}^{\mathrm{GIT}}\)
Recall from §5 that the reparametrisation group \(\hat{U}=\mathrm{Diff}_{2}(1) =\left \{u(\alpha _{1},\alpha _{2})=\right. \left. \left ( \begin{array}{c@{\quad}c} \alpha _{1} & \alpha _{2} \\ 0 & \alpha _{1}^{2} \end{array} \right ): \alpha _{1} \in {\mathbb{C} }^{*},\alpha _{2} \in {\mathbb{C} } \right \}\) acts on the projective space \({\mathbb{P} }({\mathbb{C} }\oplus \,{\mathrm{Hom}} \,({\mathbb{C} }^{2}, { \mathbb{C} }^{n}))= {\mathbb{P} }[x:f':f'']\) via
Note that \(f',f'' \in {\mathbb{C} }^{n}\) form the columns of an \(n \times 2\) matrix \(M \in \,{\mathrm{Hom}} \,({\mathbb{C} }^{2}, {\mathbb{C} }^{n})\), and \(\mathrm{Diff}_{2}(1)\) acts on the right via matrix multiplication. Thus the grading 1-parameter subgroup is \(\lambda (t)=\left ( \begin{array}{c@{\quad}c} t & 0 \\ 0 & t^{2} \end{array} \right )\), whose weight on the 1-dimensional Lie algebra \(\mathrm{Lie}(U)\) is 1. The minimal weight space \(Z_{\min} = \{[1:0:0]\}\) consists of a single point and the blow-up of \({\mathbb{P} }({\mathbb{C} }\oplus \,{\mathrm{Hom}} \,({\mathbb{C} }^{k}, { \mathbb{C} }^{n}))\) at this point is \({\tilde{\mathbb{P} }}=\mathrm{Bl}_{[1:0:0]} {\mathbb{P} }=\{([x:v_{1}:v_{2}],[w_{1}, w_{2}]):w_{1}\otimes v_{2}=w_{2}\otimes v_{1}\}\). The minimal weight space in \({\tilde{\mathbb{P} }}\) is
which sits in the exceptional divisor \(E\) for the blow-up, and \(\tilde{Z}_{\min }\simeq {\mathbb{P} }^{n-1}\) is a projective space. The maximal unipotent is
and the \(U\)-stabiliser of any point in \(\tilde{Z}_{\min }\) is trivial. Hence stability coincides with semistability for the induced \(\mathrm{Diff}_{2}(1)\) action on \({\tilde{\mathbb{P} }}\). The projection \(p: {\tilde{\mathbb{P} }}_{\min}^{0} \to \tilde{Z}_{\min }\) has affine fibres and \(U\) acts fibrewise, so \(p\) induces a fibration \(\bar{p}: X_{2}^{\mathrm{GIT}} = {\tilde{\mathbb{P} }}/\!/ \mathrm{Diff}_{2}(1) \to \tilde{Z}_{\min }\) where the fibre over \(z\) is isomorphic to a (weighted) projective space \(((p^{-1}(z)\setminus Uz)/U)/\lambda ({\mathbb{C} }^{*})\). More concretely,
Assume \(w_{1}=(w_{1}^{1},\ldots , w_{1}^{n})^{T}\) with \(w_{1}^{1} \neq 0\). Then \((p^{-1}(z)/U)\) can be identified with a slice for the \(U\) action on \(p^{-1}(z)\): for any \((\lambda w_{1}. w_{2}) \in p^{-1}(z)\) there is a unique \(\left ( \begin{array}{c@{\quad}c} 1 & \alpha _{2} \\ 0 & 1 \end{array} \right ) \in U\) such that
Hence \(p^{-1}(z)/U \simeq {\mathbb{C} }^{n}\) with coordinates \(\lambda , \tilde{w}_{2}^{2}, \ldots , \tilde{w}_{2}^{n}\). The weight of the well-adapted shift of \(\lambda ({\mathbb{C} }^{*})\) on \(\lambda \) is \(-\epsilon <0\) and \(1-\epsilon >0\) on \(\tilde{w}_{2}^{2}, \ldots , \tilde{w}_{2}^{n}\), hence \((p^{-1}(z)/U)/\!/{\mathbb{C} }^{*}= {\mathbb{P} }^{n-1}\). Thus \(X_{k}^{\mathrm{GIT}}\) fibres over \(\tilde{Z}_{\min }= {\mathbb{P} }^{n-1}\) with fibres isomorphic to \({\mathbb{P} }^{n-1}\).
Note that in this case \(N=1\), and sections of \(\mathcal{O}_{X_{2}^{\mathrm{GIT}}}(1)\) are given by coordinates on this affine \(U\)-slice. In (5.3) above
and hence the minor \(\left | \begin{array}{c@{\quad}c} w_{1}^{1} & w_{2}^{1} \\ w_{1}^{i} & w_{2}^{i} \end{array} \right |=w_{2}^{i} w_{1}^{1}-w_{1}^{i} w_{2}^{1}\) gives such a section. This is indeed invariant under \(\mathrm{Diff}_{k}(1)\), its weighted degree is \(1+2=3\), corresponding to the bundle \(E_{2,3}\) of semi-invariant jet differentials. The sections of \(\mathcal{O}_{X_{2}^{\mathrm{GIT}}}(m)\) are given by degree \(m\) polynomials in the \(2 \times 2\) minors of \((w_{1},w_{2})\).
5.2 Description of \(X_{3}^{\mathrm{GIT}}\)
When \(k=3\), the reparametrisation group
acts on the projective space \({\mathbb{P} }({\mathbb{C} }\oplus \,{\mathrm{Hom}} \,({\mathbb{C} }^{3}, { \mathbb{C} }^{n}))= {\mathbb{P} }[x:v_{1}:v_{2}:v_{3}]\) via
With \(\lambda (t)=\left ( \begin{array}{c@{\quad}c@{\quad}c} t & 0 & 0 \\ 0 & t^{2} & 0 \\ 0 & 0 & t^{3}\end{array} \right )\) the minimal weight space on the blow-up \({\tilde{\mathbb{P} }}\) is
The fibre of the projection \(p\) is
Assume \(w_{1}^{1} \neq 0\), then there is a unique \(\left ( \begin{array}{c@{\quad}c@{\quad}c} 1 & \alpha _{2} & \alpha _{3} \\ 0 & 1 & 2\alpha _{2} \\ 0 & 0 & 1\end{array} \right ) \in U\) which eliminates the first row:
Hence \(p^{-1}(z)/U \simeq {\mathbb{C} }^{2n-1}\) with coordinates \(\lambda , \tilde{w}_{2}^{2}, \ldots , \tilde{w}_{2}^{n}, \tilde{w}_{2}^{2}, \ldots , \tilde{w}_{2}^{n}\) and adapted \(\lambda ({\mathbb{C} }^{*})\)-weight vector \((-\epsilon , 1-\epsilon ,\ldots , 1-\epsilon , 2-\epsilon , \ldots , 2- \epsilon )\). Therefore \((p^{-1}(z)/U)/\!/ {\mathbb{C} }^{*}= {\mathbb{P} }[1,\ldots , 1,2, \ldots , 2]\) is a weighted projective space. Thus \(X_{3}^{\mathrm{GIT}}\) fibres over \(\tilde{Z}_{\min }= {\mathbb{P} }^{n-1}\) with fibres isomorphic to weighted projective spaces.
The sections of \(\mathcal{O}_{X_{3}^{\mathrm{GIT}}}(1)\) come again from the coordinates on the constructed affine \(U\)-slice: \(\tilde{w}_{2}^{i}\) correspond to \(2 \times 2\) minors, and the new invariants come from \(\tilde{w}_{3}^{i}\). From (5.4) one obtains
and hence
which gives the invariant section
To get all sections of powers of \(\mathcal{O}_{X_{3}^{\mathrm{GIT}}}(1)\), we run the upper index pair \((1,i)\) in this formula over all pairs \((i,j)\), \(1\le i,j \le 3\), \(i\neq j\). One can check that these are indeed invariant under \(U\) with the weighted degree \(1+1+3=2+2+1=5\), corresponding to the bundle \(E_{3,5}\) of semi-invariant jet differentials.
5.3 \(X_{k}^{\mathrm{GIT}}\) for higher \(k\)
For general \(k\geq 2\) a similar argument gives the following picture:
Proposition 5.5
\(X_{k}^{\mathrm{GIT}}\) is a fibration over \(\tilde{Z}_{\min }\simeq {\mathbb{P} }^{n-1}\) with fibres isomorphic to a weighted projective space \({\mathbb{P} }[1,\ldots , 1,2,\ldots , 2, \ldots , k-1,\ldots , k-1]\) with all weights repeated \(n-1\) times. Sections of \(\mathcal{O}_{X_{k}^{\mathrm{GIT}}}\) are given by homogenisation of coordinates on the canonical section constructed as above for \(k= 2,3\).
Note that
6 Reducing hyperbolicity to intersection theory
Now let \(X\subseteq {\mathbb{P} }^{n+1}\) be a smooth projective hypersurface of dimension \(n\) and degree \(\deg (X)=d\) in \({\mathbb{P} }^{n+1}\). Recall from Remark 5.4 that there is an integer \(N>0\) and an ample line bundle \(\mathcal{O}_{X_{k}^{\mathrm{GIT}}}(1)\) on \(X_{k}^{\mathrm{GIT}}\) which pulls back to \(L^{\otimes N}\) on \({\tilde{\mathbb{P} }}^{s,\hat{U}}\), constructed as the geometric quotient of the restriction of \(L^{\otimes N}\) to \({\tilde{\mathbb{P} }}^{s,\hat{U}}\) by the well-adapted linear action of \(\mathrm{Diff}_{k}(1)\). Here \(N\) is sufficiently divisible that for every \(x \in {\tilde{\mathbb{P} }}^{s,\hat{U}}\) the finite stabiliser \(\operatorname{Stab}_{\hat{U}}(x)\) acts trivially on the fibre of \(L^{\otimes N}\) at \(x\). The fibre \(X_{k}^{\mathrm{GIT}}\) of \(\mathcal{X}_{k}^{\mathrm{GIT}} \to X\) is a non-reductive GIT quotient \({\tilde{\mathbb{P} }}/\!/\hat{U}\) where \(\hat{U}= \mathrm{Diff}_{k}\), and is a projective variety of dimension \(\dim (X_{k}^{\mathrm{GIT}})=k(n-1)\) with at worst orbifold singularities. Moreover there is an identification
where \(\tilde{J}_{k}^{s,\hat{U}}X\) is a \(\mathrm{Diff}_{k}(1)\)-invariant open subset of the blow-up \(\tilde{J}_{k}X\) of \(J_{k}X\) along its zero section, and a relatively ample line bundle \(\mathcal{O}_{\mathcal{X}_{k}^{\mathrm{GIT}}}(1)\) on \(\mathcal{X}_{k}^{\mathrm{GIT}} \to X\), constructed as the geometric quotient of
by the action of \(\mathrm{Diff}_{k}(1)\), which pulls back on each fibre of \(\mathcal{X}_{k}^{\mathrm{GIT}} \to X\) to \(L^{\otimes N}\) (or equivalently to \(\mathcal{O}(-NE)\) where \(E\) is the exceptional divisor for the blow-up) on \({\tilde{\mathbb{P} }}^{s,\hat{U}} = {\tilde{\mathbb{P} }}^{0}_{\min} \setminus U Z_{\min}( {\tilde{\mathbb{P} }})\).
Remark 6.1
Our proof of Theorem 1.3 will involve checking that an integral over \(\mathcal{X}_{k}^{\mathrm{GIT}}\) of a cohomology class depending on \(N\) is strictly positive. However it will turn out that this integral is independent of \(N\) up to multiplication by factors of \(N\), so its positivity does not depend on \(N\).
First we need to relate the direct image sheaf \(\pi _{*}\mathcal{O}_{\mathcal{X}_{k}^{\mathrm{GIT}}}(m)\) to the Demailly–Semple bundles \(E_{k,j}\) of invariant jet differentials of order \(k\) and weighted degree \(j\).
Proposition 6.2
Let \(\pi : \mathcal{X}_{k}^{\mathrm{GIT}} \to X\) denote the projection and \(N\) be as defined as in Remarks 3.12and 5.4. If \(k>1\) then the direct image sheaf
is a subsheaf of the sheaf of holomorphic sections of \(E_{k,\le N^{2}km}=\oplus _{j=0}^{N^{2}km}E_{k,j}\).
Proof
The result to be proved is local on \(X\) so it suffices to work in local coordinates in a neighbourhood of \(x \in X\). Then the derivatives \(f'(0),\ldots , f^{(k)}(0)\) of holomorphic maps \(f: ({\mathbb{C} },0) \to (X,x)\) with respect to these local coordinates on \(X\) provide coordinates on the fibres of the affine bundle \(J_{k}X \to X\).
Recall from \(\S \)2.3 that \(E_{k,j}\) is a subbundle of the bundle
over \(X\), where \(\mathcal{E}_{k,j}\) is the space of weighted degree \(j\) complex-valued polynomials \(Q(f'(0),\ldots , f^{(k)}(0))\) on the space \(J_{k}(1,n)\) of \(k\)-jets of holomorphic maps \(f: ({\mathbb{C} },0) \to ({\mathbb{C} }^{n},0)\). Here \(\mathcal{E}_{k,j}\) has commuting left and right actions of \(\mathrm{Diff}_{k}(n)\) and \(\mathrm{Diff}_{k}(1)\), and
where \(U = U_{k}\) is the unipotent radical of \(\mathrm{Diff}_{k}(1)\) and \(\hat{U}= \mathrm{Diff}_{k}(1)\).
Recall also from Definition 5.3 that
where \(X_{k}^{\mathrm{GIT}} = {\tilde{\mathbb{P} }}^{s,\hat{U}}/\hat{U}\) is a non-reductive GIT quotient which is a projective completion of \(J_{k}^{\text{reg}}(1,n)/\mathrm{Diff}_{k}(1)\). Here \({\tilde{\mathbb{P} }}^{s,\hat{U}}\) is an open subvariety of the blow-up of the affine space \(J_{k}(1,n)\cong \,{\mathrm{Hom}} \,({\mathbb{C} }^{k}, {\mathbb{C} }^{n})\) at the origin. The action of \(\mathrm{Diff}_{k}(n)\) on \(J_{k}(1,n)\) is not linear but it fixes the origin and hence acts on this blow-up. It preserves \({\tilde{\mathbb{P} }}^{s,\hat{U}}\) and commutes with the action of \(\mathrm{Diff}_{k}(1)\), so there is an induced action of \(\mathrm{Diff}_{k}(n)\) on the geometric quotient \(X_{k}^{\mathrm{GIT}}= {\tilde{\mathbb{P} }}^{s,\hat{U}}/\hat{U}\).
By Remark 5.4, a section \(\sigma \) of \(\bigoplus _{j=1}^{m} \mathcal{O}_{\mathcal{X}_{k}^{\mathrm{GIT}}}(j)\) pulls back over each fibre of \(\mathcal{X}_{k}^{\mathrm{GIT}} \to X\) to the restriction of a \(\mathrm{Diff}_{k}(1)\)-invariant section (with respect to the well-adapted linearisation) \(\tilde{\sigma }\) of \(\bigoplus _{j=1}^{m} \mathcal{O}(-NjE)\) over the blow-up of \(J_{k} X\) along its zero section, where \(E\) is the exceptional divisor for this blow-up and \(\tilde{\sigma }\) is polynomial of degree at most \(Nm\) in the coordinates \(f'(0),\ldots , f^{(k)}(0)\) on the corresponding fibre of \(J_{k}X \to X\), when we use the local coordinates on \(X\) to identify the \(j\)th column of \(J_{k}(1,n)\cong \,{\mathrm{Hom}} \,({\mathbb{C} }^{k}, {\mathbb{C} }^{n})\) with the \(j\)th derivative \(f^{(j)}(0)\) of a holomorphic jet \(f\) at \(x\). Then \(\tilde{\sigma }\) has the form
where \(Q_{i}\) is a homogeneous polynomial of degree \(i\) in its entries \(f'(0),\ldots , f^{(k)}(0)\) and is invariant under the well-adapted action of \(\mathrm{Diff}_{k}(1)\). For this well-adapted action the weighted degree of \(f^{(j)}(0)\) is \(jN-\chi \) for some \(\chi >0\), determined by the well-adapted shift of \(L^{\otimes N}\). Well-adaptedness means that
and hence \(N<\chi \ll 2N\) is slightly bigger than \(N\). A well-adapted weighted-homogeneous degree \(d\) monomial \((f')^{i_{1}} \ldots (f^{(k)})^{i_{k}}\) of degree \(i_{1} + \cdots + i_{k}\) satisfies
and hence \(N(i_{1}+2i_{2}+\cdots +ki_{k})=d+\chi (i_{1}+\cdots +i_{k})\); if this monomial appears with nonzero coefficient in \(Q_{i}\) then \(i_{1} + \cdots + i_{k} = i \le Nm\) and by invariance \(d=0\). Hence \(Q_{i}\) is weighted homogeneous for the non-shifted linearisation \(L^{\otimes N}\) of weighted degree \(N(i_{1}+2i_{2}+\cdots +ki_{k})= \chi (i_{1}+\cdots +i_{k}) < 2N^{2} m \le N^{2} km\) as required. □
The following classical theorem connects global invariant jet differentials to the GGL conjecture.
Theorem 6.3
Fundamental vanishing theorem, Green–Griffiths [27], Demailly [19], Siu [39]
Let \(N\) be the integer defined as in Remarks 3.12and 5.4. Assume that there exist integers \(k,m>0\) and ample line bundle \(A\to X\) such that there are nonzero global sections
Let \(\sigma _{1},\ldots ,\sigma _{\ell}\) be arbitrary nonzero sections and let \(Z\subseteq J_{k}X\) be the base locus of these sections. Then every entire holomorphic curve \(f:{\mathbb{C} }\to X\) necessarily satisfies \(f_{[k]}({\mathbb{C} })\subseteq Z\). In other words, for every global \(\mathrm{Diff}_{k}(1)\)-(semi)invariant differential equation \(P\) vanishing on an ample divisor, every entire holomorphic curve \(f\) must satisfy the algebraic differential equation \(P(f'(t),\ldots , f^{(k)}(t))\equiv 0\).
By Diverio [23, Theorem 1], for arbitrary ample \(A\), \(H^{0}(X,E_{k,m} \otimes A^{-1})= 0\) holds for all \(m\ge 1\) if \(k< n\), so we can restrict our attention to the range \(k\ge n\). It turns out that \(k=n\) is a good choice; in particular Theorem 6.4 below, which is a crucial ingredient for our proof of Theorem 1.3, holds when \(k=n\).
To control the order of vanishing of these differential forms along the ample divisor we will choose \(A\) to be (as in [25]) a proper twist of the canonical bundle of \(X\). Recall that the canonical bundle of the smooth, degree \(d\) hypersurface \(X\) is
which is ample as soon as \(d\ge n+3\). The following theorem summarises the results of §3 in Diverio–Merker–Rousseau [25], using the improved linear pole order for slanted vector fields due to Darondeau [18].
Theorem 6.4
Algebraic degeneracy of entire curves [25] and [18]
Assume that \(k=n\), that \(N \geq 1\) is chosen as in Remarks 3.12and 5.4, and that there exist \(\delta =\delta (n) >0\) and \(M=M(n,\delta )\) such that
whenever \(\deg (X)>M(n,\delta )\) and \(m \gg 0\). Then the Green-Griffiths-Lang conjecture holds whenever
We will prove the following theorem.
Theorem 6.5
Let \(X\subseteq {\mathbb{P} }^{n+1}\) be a smooth complex hypersurface of degree \(d\) with ample canonical bundle; that is \(d \ge n+3\). Let \(N \ge 1\) be as in Remarks 3.12and 5.4. Then
provided that (i) \(\delta =\frac{1}{16n^{3}}\), (ii) \(\deg (X)>16n^{3}(5n+4)\) and (iii) \(2\delta Nnm\) is an integer and is sufficiently large.
Theorem 1.3 will follow from Theorem 6.4 once we have proved Theorem 6.5.
To prove Theorem 6.5 we will use the algebraic Morse inequalities of Demailly and Trapani to reduce the existence of global sections to the positivity of certain tautological integrals over \(\mathcal{X}_{n}^{\mathrm{GIT}}\). Let \(L\to P\) be a holomorphic line bundle over a compact Kähler manifold \(P\) of dimension \(p\) and let \(E \to P\) be a holomorphic vector bundle of rank \(r\).
Theorem 6.6
Algebraic Morse inequalities, Demailly [20], Trapani [43]
Suppose that \(L=F\otimes G^{-1}\) is the difference of two nef line bundles \(F\), \(G\) on \(P\). Then for any nonnegative integer \(q\in {\mathbb{Z} }_{\ge 0}\)
In particular, the case when \(q=1\) asserts that \(L^{\otimes m}\otimes E\) has a nonzero global section for \(m\) large provided that the intersection number \(F^{p}-pF^{p-1}G\) is strictly positive.
In order to apply this theorem we need an expression for \(\mathcal{O}_{\mathcal{X}_{n}^{\mathrm{GIT}}}(1)\otimes \pi ^{*}K_{X}^{-2 \delta Nn}\) as a difference of nef bundles on \(\mathcal{X}_{n}^{\mathrm{GIT}}\).
Proposition 6.7
Let \(d\ge n+3\) so that \(K_{X}\) ample. Let \(N\ge 1\) be fixed as in Remarks 3.12and 5.4. The following line bundles are nef on \(\mathcal{X}_{n}^{\mathrm{GIT}}\):
-
1.
\(\mathcal{O}_{\mathcal{X}_{n}^{\mathrm{GIT}}}(1) \otimes \pi ^{*} \mathcal{O}_{X}(2N)\)
-
2.
\(\pi ^{*}\mathcal{O}_{X}(2N)\otimes \pi ^{*}K_{X}^{2\delta Nn}\) for any \(\delta >0\) and \(2\delta Nn\) integer.
Remark 6.8
Since \(K_{X}=\mathcal{O}_{X}(d-n-2)\) we can write \(\pi ^{*}\mathcal{O}_{X}(2N)\otimes \pi ^{*}K_{X}^{2\delta Nn}\) as \(\pi ^{*}\mathcal{O}_{X}(2N + 2\delta N n(d-n-2))\). However we prefer to keep the terms \(\pi ^{*}\mathcal{O}_{X}(2N)\) and \(\pi ^{*}K_{X}^{2\delta Nn}\) separate.
First note that \(T^{*}_{ {\mathbb{P} }^{n+1}}\otimes \mathcal{O}(2)\) is globally generated, and there is a surjective bundle map \((T^{*}_{ {\mathbb{P} }^{n+1}}\otimes \mathcal{O}(2))|_{X} \rightarrow T^{*}_{X}\otimes \mathcal{O}_{X}(2)\). Therefore
We now follow an inductive argument. Eliminating the terms of degree \(k+1\) results in a surjective algebra homomorphism \(J_{k}(1,n) \twoheadrightarrow J_{k-1}(1,n))\), and the chain \(J_{k}(1,n) \twoheadrightarrow J_{k-1}(1,n) \twoheadrightarrow \cdots \twoheadrightarrow J_{1}(1,n)\) induces an increasing filtration on \(J_{k}(1,n)^{*}\):
which gives a short exact sequence of bundles
This induces a short exact sequence of tangent bundles at the zero section
Note that the tangent bundles are vector bundles, and hence this is a short exact sequence of vector bundles over \(X\). By induction and (6.1) we get that \(T_{0}(J_{k}X^{*}) \otimes \mathcal{O}(2k)\) is globally generated, and hence
is relatively nef line bundle (note that in this paper we set \({\mathbb{P} }(V)\) to be the bundle of 1-dimensional subspaces in \(V\), not the 1-dimensional quotients, hence sections of \(\mathcal{O}_{ {\mathbb{P} }(V)}(1)\) are points in \(V^{*}\)).
By Remark 5.4 the relatively ample line bundle \(\mathcal{O}_{\mathcal{X}_{k}^{\mathrm{GIT}}}(1)\) on \(\mathcal{X}_{k}^{\mathrm{GIT}} \to X\), constructed as the geometric quotient of
by the action of \(\mathrm{Diff}_{k}(1)\), pulls back on each fibre of \(\mathcal{X}_{k}^{\mathrm{GIT}} \to X\) to \(L^{\otimes N}\) (or equivalently to \(\mathcal{O}(-NE)\) where \(E\) is the exceptional divisor for the blow-up) on \({\tilde{\mathbb{P} }}^{s,\hat{U}} = {\tilde{\mathbb{P} }}^{0}_{\min} \setminus U Z_{\min}( {\tilde{\mathbb{P} }})\). Hence (6.4) tells us that \(L^{\otimes N} \otimes \tilde{\pi}^{*}\mathcal{O}(2N)\) is nef, and the induced bundle \(\mathcal{O}_{\mathcal{X}_{n,\mathrm{GL}}^{\mathrm{GIT}}}(1) \otimes \pi ^{*}\mathcal{O}_{X}(2N)\) is also nef. The second part follows from the standard fact that the pull-back of an ample line bundle is nef. □
Consequently, we can express \(\mathcal{O}_{\mathcal{X}_{n}^{\mathrm{GIT}}}(1)\otimes \pi ^{*}K_{X}^{-2 \delta nN}\) as the following difference of two nef line bundles:
We will be able to deduce Theorem 6.5 from the algebraic Morse inequalities Theorem 6.6 by proving that the top degree form \(I(n,\delta )\) on \(\mathcal{X}_{n}^{\mathrm{GIT}}\) given by
is positive, in the sense that
if \(\delta =\frac{1}{16n^{3}}\) and \(d>N(n,\delta )=16n^{3}(5n+4)\).
7 Cohomology of \(X_{k}^{\mathrm{GIT}}= {\tilde{\mathbb{P} }}/\!/\mathrm{Diff}_{k}(1)\)
Recall that the fibre \(X_{k}^{\mathrm{GIT}}\) of \(\mathcal{X}_{k}^{\mathrm{GIT}} \to X\) is the non-reductive GIT quotient \(X_{k}^{\mathrm{GIT}} = {\tilde{\mathbb{P} }}/\!/ \mathrm{Diff}_{k}(1)\) of the blow-up \({\tilde{\mathbb{P} }}\) of a point in the projective space \({\mathbb{P} }({\mathbb{C} }\oplus \,{\mathrm{Hom}} \,({\mathbb{C} }^{k},{ \mathbb{C} }^{n}))\) by the action of \(\mathrm{Diff}_{k}(1) = \hat{U}= U \rtimes \lambda ({\mathbb{C} }^{*})\). Our aim is to express \(\int _{\mathcal{X}_{n}^{\mathrm{GIT}}} I(n,\delta ) \) as an integral over \(X\) by representing \(I(n,\delta )\) using an equivariant cohomology class and integrating over the fibres \(X_{k}^{\mathrm{GIT}}\) of \(\mathcal{X}_{k}^{\mathrm{GIT}} \to X\). Recall also that since \(U\) is contractible, \(\hat{U}\) is homotopy equivalent to \({\mathbb{C} }^{*}\) and to \(S^{1}\), and so \(\hat{U}\)-equivariant cohomology is isomorphic to \({\mathbb{C} }^{*}\)- (or equivalently \(S^{1}\)-) equivariant cohomology of \({\tilde{\mathbb{P} }}\).
Recall also from Theorem 4.7 and Remark 4.8 that we can integrate a cohomology class on the nonreductive GIT quotient \(X_{k}^{\mathrm{GIT}} = {\tilde{\mathbb{P} }}/\!/\hat{U}\) by expressing it as the image \(\kappa _{\hat{U}} (\eta )\) under the surjective ring homomorphism \(\kappa _{\hat{U}}: H_{\hat{U}}^{*}( {\tilde{\mathbb{P} }};{\mathbb{Q} })=H_{S^{1}}^{*}( {\tilde{\mathbb{P} }};{\mathbb{Q} }) \to H^{*}( {\tilde{\mathbb{P} }}/\!/ \hat{U};{\mathbb{Q} })\) of some \(\hat{U}\)-equivariant cohomology class \(\eta \) on \({\tilde{\mathbb{P} }}\) represented by an equivariant differential form \(\eta (z)\), where \(z\) is the standard coordinate on the Lie algebra of \(\lambda ({\mathbb{C} }^{*})\); we have
where \(F_{\min} = Z_{\min}( {\tilde{\mathbb{P} }})\) is the union of those connected components of the fixed point locus \({\tilde{\mathbb{P} }}^{{\mathbb{C} }^{*}}\) on which the \(S^{1}\)-moment map takes its minimum value \(\omega _{\min}\), and \(n_{\hat{U}}\) is a strictly positive rational number which depends only on \(\hat{U}\) and the size of the stabiliser in \(\hat{U}\) of a generic \(x \in X\). Here \(\mathcal{N}_{UF_{\min}}\) is the normal bundle to \(UF_{\min}\) in \({\tilde{\mathbb{P} }}\), and in order to calculate the residue we express the restriction to \(F_{\min} \) of its equivariant Euler class as a polynomial in \(z\) with coefficients in the cohomology of \(F_{\min}\), all but one of degree at least one and thus nilpotent, so that the multiplicative inverse can be expressed as a Laurent series in \(z\) with coefficients in the cohomology of \(F_{\min}\). This means that it is in fact more natural to write this formula as
Moreover as noted in Remark 4.8, it suffices to express our cohomology class on \({\tilde{\mathbb{P} }}/\!/\hat{U}\) as
where \(\tilde{J}_{k}(1,n)\) is the blow-up of \(J_{k}(1,n)\) at 0 and
Thus to understand the cohomology of \(X_{k}^{\mathrm{GIT}}\) we want to consider first the equivariant cohomology of \(Z_{\min}( {\tilde{\mathbb{P} }}) \subseteq \tilde{J}_{k}(1,n) \subseteq {\tilde{\mathbb{P} }}\) with respect to the action of \(\mathrm{Diff}_{k}(1) = \hat{U}= U \rtimes \lambda ({\mathbb{C} }^{*})\).
Since \(J_{k}(1,n)\) is an affine space, \(\tilde{J}_{k}(1,n)\) retracts equivariantly onto its exceptional divisor \(E \cong {\mathbb{P} }^{kn-1}\), which is a GIT quotient of \(T_{0} J_{k}(1,n)\) by the \({\mathbb{C} }^{*}\)-action of scalar multiplication which commutes with the action of \(\hat{U}\). Let \(T_{2}\) be the two-dimensional complex torus which is the product of this \({\mathbb{C} }^{*}\) acting via scalar multiplication and the one-parameter subgroup \(\lambda :{\mathbb{C} }^{*} \to \hat{U}\) of \(\hat{U}\); its Lie algebra has coordinates \((w,z)\) where \(w\) is the standard coordinate on the Lie algebra of the copy of \({\mathbb{C} }^{*}\) acting via scalar multiplication and \(z\) is the standard coordinate on the Lie algebra of \(\lambda ({\mathbb{C} }^{*})\). The \(T_{2}\)-weights on the tangent space \(T_{0} J_{k}(1,n) = T_{[1:0:\ldots :0]} {\mathbb{P} }({\mathbb{C} } \oplus \,{\mathrm{Hom}} \,({\mathbb{C} }^{k}, {\mathbb{C} }^{n}))\) are \(w+z,w+2z,\ldots , w+kz\) (all of these occurring with multiplicity \(n\)).
Embedding \({\tilde{\mathbb{P} }}\) in \({\mathbb{P} }^{kn} \times {\mathbb{P} }^{kn-1}\), let \(\zeta \) denote the equivariant first Chern class of the hyperplane line bundle on \(P^{kn}\) (which restricts to the trivial bundle on \(J_{k}(1,n)\)), and let \(\eta \) denote the equivariant first Chern class of the hyperplane line bundle on \({\mathbb{P} }^{kn-1} = {\mathbb{P} }(T_{0} J_{k}(1,n))\), with both pulled back to equivariant classes on \({\tilde{\mathbb{P} }}\). To apply Theorem 3.6 we take \(a\gg 1\) and twist the restriction to \({\tilde{\mathbb{P} }}\) of the ample linearisation \(L=\mathcal{O}_{ {\mathbb{P} }^{kn}}(a) \otimes \mathcal{O}_{ { \mathbb{P} }^{kn-1}}(1)\) on \({\mathbb{P} }^{kn} \times {\mathbb{P} }^{kn-1}\) by a rational character of \(\hat{U}\) which can be identified with \(bz\) for some \(b \in {\mathbb{Q} }\) and makes the linearisation well-adapted for the action of \(\hat{U}\). Then \(\tilde{Z}_{\min }= Z_{\min}( {\tilde{\mathbb{P} }})\) is the \(n-1\)-dimensional projective linear subspace of \(E = {\mathbb{P} }(T_{0}J_{k}(1,n) \cong {\mathbb{P} }^{kn-1}\) corresponding to the \(T_{2}\)-weight space in \(T_{0} J_{k}(1,n)\) with weight \(w+z\), and to make the linearisation well-adapted for the \(\hat{U}\)-action we twist by a rational character of \(\hat{U}\) (and thus of \(T_{2}\)) so that this weight \(w+z\) becomes \(w-\epsilon z\) for some small positive \(\epsilon \). So the restriction of the well-adapted \(\hat{U}\)-equivariant linearisation on \({\tilde{\mathbb{P} }}\) to \(\tilde{Z}_{\min }= Z_{\min}( {\tilde{\mathbb{P} }})\) can be identified (up to replacement with a positive tensor power) with the ample \(\hat{U}\)-equivariant line bundle (defined up to replacement with a positive tensor power) induced on \(Z_{\min}( {\tilde{\mathbb{P} }})\) by regarding \(Z_{\min}( {\tilde{\mathbb{P} }})\) as the GIT quotient by \({\mathbb{C} }^{*}\) of the \(n\)-dimensional weight space in \(T_{0} J_{k}(1,n)\) with \({\mathbb{C} }^{*} \times \hat{U}\)-weight \(w -\epsilon z\) where \(0 < \epsilon \ll 1\). In fact in §5.2 of [9] it is shown that, because the action of \(\hat{U}\) on \({\tilde{\mathbb{P} }}\) extends to an action of \(\mathrm{GL}(k)\), we can choose any rational \(0<\epsilon <1\); we will make a choice of \(\epsilon \) in this range later.
Finally recall that the non-reductive GIT quotient \(X_{k}^{\mathrm{GIT}}= {\tilde{\mathbb{P} }}/\hat{U}= { \tilde{\mathbb{P} }}/\!/\mathrm{Diff}_{k}\) given by Theorem 3.6 is a smooth projective variety of dimension \(\dim (X_{k}^{\mathrm{GIT}})=k(n-1)\), and a sufficiently divisible power
of \(L\) induces an ample line bundle \(\mathcal{O}_{X_{k}^{\mathrm{GIT}}}(1)\) on this quotient; this line bundle pulls back to the restriction of \(L^{\otimes N}\) to the (semi)stable locus in \({\tilde{\mathbb{P} }}\) (see Remark 3.7). The first Chern class of \(\mathcal{O}_{X_{k}^{\mathrm{GIT}}}(1)\) has an equivariant lift to \({\tilde{\mathbb{P} }}\) which is the first Chern class of the well-adapted twist of \(L^{\otimes N}\), and by choosing \(N\) to be sufficiently divisible we can assume that its restriction to \(\tilde{Z}_{\min }= Z_{\min}( {\tilde{\mathbb{P} }})\) is the ample \(\hat{U}\)-equivariant line bundle induced on \(Z_{\min}( {\tilde{\mathbb{P} }})\) by regarding \(Z_{\min}( {\tilde{\mathbb{P} }})\) as the GIT quotient by \({\mathbb{C} }^{*}\) of the \(n\)-dimensional weight space \(T_{0} J_{k}(1,n)_{w+z}\) in \(T_{0} J_{k}(1,n)\) with \({\mathbb{C} }^{*} \times \hat{U}\)-weight \(N(w -\epsilon z)\).
Our aim is to show that the integral \(\int _{\mathcal{X}_{n}^{\mathrm{GIT}}} I(n,\delta ) \) is strictly positive by integrating over the fibres of \(\mathcal{X}_{n}^{\mathrm{GIT}} \to X\) to obtain an integral over \(X\). We will find that in order to integrate over the fibres we will need to calculate
when \(k=n\), as well as some other similar integrals. It follows from (7.2) that
By applying (7.2) in the case when \(U\) is trivial and \(F_{\min} = \{0\} \subseteq T_{0} J_{k}(1,n)_{w+z} \subseteq { \mathbb{P} }({\mathbb{C} }\oplus T_{0} J_{k}(1,n))\), we have
where \(\gamma (w,z)\) is a \(T_{2}\)-equivariant cohomology class on \(T_{0} J_{k}(1,n)_{w+z}\) which represents the restriction of the equivariant first Chern class of the well-adapted twist of \(L^{\otimes N}\) to \(Z_{\min}( {\tilde{\mathbb{P} }}) = {\mathbb{P} }(T_{0} J_{k}(1,n)_{w+z})\) quotiented by the equivariant Euler class \(\mathrm{Euler}(\mathcal{N}_{U\tilde{Z}_{\min }})(z)\), and \(\mathcal{N}_{T_{0} J_{k}(1,n)_{w+z} }\) is the normal bundle to \({T_{0} J_{k}(1,n)_{w+z} }\) in \(T_{0}J_{k}(1,n)\). Note that the normal bundle \(\mathcal{N}_{U\tilde{Z}_{\min }}\) to \(U \tilde{Z}_{\min }\) in \({\tilde{\mathbb{P} }}\) is the quotient of the normal bundle \(\mathcal{N}_{\tilde{Z}_{\min }}\) to \(\tilde{Z}_{\min }\) by the trivial bundle \(V_{{\mathfrak {u}}}\) with fibre \({\mathfrak {u}}\).
Thus we will obtain an expression for \(\int _{X_{k}^{\mathrm{GIT}}} c_{1}(\mathcal{O}_{X_{k}^{\mathrm{GIT}}}(1))^{k(n-1)}\) as an iterated residue of a rational function of \(w\) and \(z\).
7.1 Integration on \(X_{k}^{\mathrm{GIT}}\)
Let us collect the ingredients needed to calculate \(\int _{X_{k}^{\mathrm{GIT}}} \,\, c_{1}(\mathcal{O}_{X_{k}^{ \mathrm{GIT}}}(1))^{k(n-1)} \) using Theorem 4.7 and Remark 4.8 as above. We have fixed a linearisation for the action of \(\hat{U}= \mathrm{Diff}_{k}(1)\) on \({\tilde{\mathbb{P} }}\) with induced ample line bundle \(\mathcal{O}_{X_{k}^{\mathrm{GIT}}}(1)\) on \(X_{k}^{\mathrm{GIT}}={ {\tilde{\mathbb{P} }}/\!/\mathrm{Diff}_{k}(1)}\) (where \({\tilde{\mathbb{P} }}\) is the projective space \({\mathbb{P} }({\mathbb{C} }\oplus \,{\mathrm{Hom}} \,({\mathbb{C} }^{k}, { \mathbb{C} }^{n}))\) blown up at \([1:0 \cdots 0]\)), and a two-dimensional complex torus \(T_{2} = {\mathbb{C} }^{*} \times \lambda ({\mathbb{C} }^{*})\) with the first copy of \({\mathbb{C} }^{*}\) acting on the tangent space \(T_{0} J_{k}(1,n)\) as scalar multiplication, and \(\lambda ({\mathbb{C} }^{*})\) acting on \(T_{0} J_{k}(1,n)\) as the restriction of the induced action of \(\hat{U}\). We have seen that
and
where \(\gamma (w,z)\) is a \(T_{2}\)-equivariant cohomology class on \(T_{0} J_{k}(1,n)_{w+z}\) which represents the restriction of the equivariant first Chern class of the well-adapted twist of \(L^{\otimes N}\) to \(\tilde{Z}_{\min }= {\mathbb{P} }(T_{0} J_{k}(1,n)_{w+z})\) quotiented by
and \(\mathcal{N}_{T_{0} J_{k}(1,n)_{w+z} }\) is the normal bundle to \({T_{0} J_{k}(1,n)_{w+z} }\) in \(T_{0}J_{k}(1,n)\). Here \(w\), \(z\) are the standard coordinates on the Lie algebra of \(T_{2}\). Substituting we obtain
The ingredients appearing in this equation can be summarised as follows.
-
1.
\(n_{\hat{U}\times {\mathbb{C} }^{*}}\) is a strictly positive rational number which depends on \({\hat{U}\times {\mathbb{C} }^{*}}\) and the size of a generic stabiliser for its action on \(T_{0} J_{k}(1,n)\).
-
2.
The restriction of the equivariant first Chern class of the well-adapted twist of \(L^{\otimes N}\) to \(\tilde{Z}_{\min }\) is represented by the \(T_{2}\)-equivariant Euler class \(Nw - N\epsilon z\) on the tangent space \(T_{0}J_{k}(1,n)\).
-
3.
The weights of \(\lambda : {\mathbb{C} }^{*} \to \hat{U}= \mathrm{Diff}_{k}(1)\) on \({\mathfrak {u}}=\mathrm{Lie}(U)\) are \(1,2,\ldots , (k-1)\), and the \(\lambda ({\mathbb{C} }^{*})\)-equivariant Euler class \(\mathrm{Euler}(V_{{\mathfrak {u}}}^{*})\) is represented by the \(T_{2}\)-equivariant class \(-z\cdot (-2z) \cdot \ldots \cdot (-(k-1)z)=(k-1)!(-z)^{k-1}\) on \(T_{0} J_{k}(1,n)\).
-
4.
The normal bundle to \(\tilde{Z}_{\min }\) in \(E\) has equivariant Euler class represented by \((-z)^{n}(-2z)^{n} \cdot \ldots \cdot (-(k-1)z)^{n}\). The weights here correspond to the second, third, …, and last columns of \(E\cong {\mathbb{P} }(T_{0}J_{k}(1,n))\). The normal bundle of \(E\) in \({\tilde{\mathbb{P} }}\) is the tautological bundle \(\mathcal{O}_{E}(-1)\) with equivariant first Chern class represented by \(-w-z\), and hence the equivariant Euler class \(\mathrm{Euler}(\mathcal{N}_{\tilde{Z}_{\min }})\) of the normal bundle \(\mathcal{N}_{\tilde{Z}_{\min }}\) to \(\tilde{Z}_{\min }\) in \({\tilde{\mathbb{P} }}\) is represented by
$$ (-1)^{(k-1)n+1}(w+z)((k-1)!)^{n}z^{(k-1)n} $$ -
5.
The equivariant Euler class \(\mathrm{Euler}(T_{0} J_{k}(1,n)_{w+z})\) is represented by \((w+z)^{n}\).
Substituting this into the formula (7.4), and changing \(z\) to \(-z\), which changes the sign of the iterated residue, we obtain the following result.
Proposition 7.1
With the notation introduced above we have
where \(n_{\hat{U}\times {\mathbb{C} }^{*}}\) is a strictly positive rational, and the iterated residue means expansion on a contour where \(|w|\gg |z|\) and taking the coefficient of \((zw)^{-1}\).
Note that the rational expression on the right hand side has positive expansion on the contour \(|w|\gg |z|\), that is, as a Laurent series in \(\frac{z}{w}\). In particular we see that the residue, which is the coefficient of \((zw)^{-1}\), is positive, as it must be, because the degree of an ample line bundle is positive.
8 Integration on \(\mathcal{X}_{k}^{\mathrm{GIT}} = \widetilde{J_{k}}X/\!/\mathrm{Diff}_{k}(1)\)
To evaluate the integral \(\int _{\mathcal{X}_{k}^{\mathrm{GIT}}}\mathcal{O}_{\mathcal{X}_{k}^{ \mathrm{GIT}}}(1)^{n+k(n-1)}\) (or the integral over \(\mathcal{X}_{k}^{\mathrm{GIT}}\) of the product of \(c_{1}(\mathcal{O}_{\mathcal{X}_{k}^{\mathrm{GIT}}}(1))^{n+k(n-1)-j}\) and the pullback to \(\mathcal{X}_{k}^{\mathrm{GIT}}\) of a cohomology class \(\zeta \) on \(X\) of degree \(j\)), we can first integrate (push forward) along the fibres of \(\pi : \mathcal{X}_{k}^{\mathrm{GIT}} \to X\) and then integrate over \(X\). Integration along the fibres can be done using the iterated residue formula of Proposition 7.1. Here we must replace \(\tilde{Z}_{\min }\) with
which we can identify with the projectivised tangent bundle \({\mathbb{P} }(TX)\) for \(X\). In the relative version of Proposition 7.1 the \(T_{2}\)-equivariant Euler class of the tangent space \(T_{0} J_{k}(1,n)_{w+z}\) is replaced with the equivariant Euler class of this tangent bundle. This is given by
where \(\lambda _{1},\ldots , \lambda _{n}\) are the Chern roots of \(TX\). Similarly the \(T_{2}\)-equivariant class representing the equivariant Euler class \(\mathrm{Euler}(\mathcal{N}_{\tilde{Z}_{\min }})\) of the normal bundle \(\mathcal{N}_{\tilde{Z}_{\min }}\) to \(\tilde{Z}_{\min }\) is replaced with
Hence after substituting \(-z\) for \(z\) as before, so that the iterated residue changes sign, we find that the relative version of Proposition 7.1 is as follows (with a similar formula for the integral over \(\mathcal{X}_{k}^{\mathrm{GIT}}\) of the product of \(c_{1}(\mathcal{O}_{\mathcal{X}_{k}^{\mathrm{GIT}}}(1))^{n+k(n-1)-j}\) and the pullback to \(\mathcal{X}_{k}^{\mathrm{GIT}}\) of a cohomology class \(\zeta \) on \(X\) of degree \(j\)).
Proposition 8.1
Here \(n_{\hat{U}\times {\mathbb{C} }^{*}}\) is a strictly positive rational, and the iterated residue means expansion on a contour where \(|z|\gg |w|\), and taking the coefficient of \((zw)^{-1}\).
The inverse of the total Chern class \(c_{X}\) of \(TX\) is the total Segre class \(S(X)\) of the tangent bundle, so \(c(X)^{-1}=s(X)\) and we can write
Hence the relative version of Proposition 7.1 gives us the following integration formula on \(\mathcal{X}_{k}^{\mathrm{GIT}}\).
Theorem 8.2
The integral over \(\mathcal{X}_{k}^{\mathrm{GIT}}\) of the product of \(c_{1}(\mathcal{O}_{\mathcal{X}_{k}^{\mathrm{GIT}}}(1))^{n+k(n-1)-j}\) and the pullback to \(\mathcal{X}_{k}^{\mathrm{GIT}}\) of a cohomology class \(\zeta \) on \(X\) of degree \(j\) is given by
Here \(n_{\hat{U}\times {\mathbb{C} }^{*}}\) is a strictly positive rational, and the residue is a homogeneous polynomial of degree \(n=\dim (X)\) in the Segre classes of \(X\), which is then integrated over \(X\).
Remark 8.3
There are three key features of this formula which makes intersection calculation significantly easier:
-
1.
It separates the Segre classes from the residue variables, and hence the residue is a polynomial of degree \(n\) in the Segre classes.
-
2.
The rational expression has positive expansion in \(\frac{z}{w}\), that is, all coefficients in the expansion on the contour \(|w|\gg |z|\) are positive.
-
3.
Although \(N\) plays a crucial role in finding the relative NEF bundles in the Morse inequalities, it only appears as a factor in the formula of Theorem 8.2, and hence the positivity of the integral does not depend on the actual value of \(N\). This is a crucial aspect of our integral formula, because we do not know the exact value of \(N\): it only guarantees the relative ampleness of the bundle \(L\).
9 Proof of Theorem 1.3
In this section we complete the proof of Theorem 6.5.
Theorem 6.5 will follow from the Morse inequalities by proving that the following top degree form on \(\mathcal{X}_{n}^{\mathrm{GIT}}\) is positive if \(\delta =\frac{1}{16n^{5}}\) and \(d>N(n,\delta )=2(4n)^{5}\):
Recall the notation \(h=c_{1}(\mathcal{O}_{X}(1))\), \(u=c_{1}(\mathcal{O}_{\mathcal{X}_{k, \mathrm{GL}}^{\mathrm{GIT}}}(1))\), and \(c_{1}=c_{1}(T_{X})\) for the corresponding first Chern classes. Then \(c_{1}(K_{X})=-c_{1}=(d-n-2)h\), and by dropping \(\pi ^{*}\) from our formula (9.1) can be rewritten as
In the residue formula of Theorem 8.2 we substitute \(u=Nw+N\epsilon z\) and we obtain
Since \(X\subseteq {\mathbb{P} }^{n+1}\) is a projective hypersurface we can express the Segre classes in Theorem 8.2 using that the Chern classes of \(X\) are expressible with \(d=\deg (X)\) and \(h\):
where \(c(X)=c(T_{X})\) is the total Chern class of \(X\). This gives
Proposition 9.1
The integral \(\int _{\mathcal{X}_{n}^{\mathrm{GIT}}} I_{n,\delta }\) is given by \(n_{\hat{U}\times {\mathbb{C} }^{*}}\) times
where
and
9.1 Residue integral formula on \(X_{2}^{\mathrm{GIT}}\)
Before we begin the analysis of the integral \(\int _{\mathcal{X}_{n}^{\mathrm{GIT}}} I_{n,\delta }\), we will study it in the case \(k=n=2\), building on the description of \(X_{2}^{\mathrm{GIT}}\) in §5.1.
For \(k=n=2\) we have \(n_{\hat{U}\times {\mathbb{C} }^{*}}=1\) and Proposition 9.1 gives
where (after dropping the \(N^{4}\) factor)
The iterated residue will not involve any power \(h^{t}\) for \(t>2\), and moreover, we cannot take terms where the degree of \(w\) in the denominator is 6 or larger. This leads to significant simplification:
The leading coefficient must be positive, which forces the inequality
and by the condition \(0< \epsilon <1\) this results in \(\delta < 1/14\), hence the first bound in Theorem 6.4 for the degree satisfies
and we can not expect better degree bound with our approach. By choosing \(\epsilon =1/4\), as in the general argument in the next sextion, the polynomial has the form
We pick \(\delta =\frac{1}{16}\) to ensure the positivity of the leading coefficient. Note that in the main theorem our choice is \(\delta =\frac{1}{16n^{3}}=\frac{1}{128}\), which is slightly smaller; this is due to estimation of the leading coefficient. Then the integral is positive for \(d>99+\sqrt{9370} \sim 200\). Hence by Theorem 6.4 the Green-Griffiths-Lang conjecture holds for
With fine-tuning the parameters we can get slighlty closer to the optimal bound \(d=186\) mentioned above.
Remark 9.2
Our approach can not compete with the best known degree bounds for small \(n\). In particular, for projective surfaces \(X \subset {\mathbb{P} }^{3}\) the best known degree bound is \(\deg (X)=18\) (see Sect. 5.4 in [24]), and we do not have a well-established explanation for the gap between our result and the best bound. Intuitively, due to well-adaptedness and the ample twist determined by \(\epsilon \) and \(N\), our non-reductive GIT model does sees only a graded sub-algebra of the invariant jet-differentials.
9.2 A first look at the iterated residue formula
As a first step in the analysis of the residue formula of Proposition 9.1 we write this integral as a polynomial in \(d\) and study its leading coefficient. For a nonnegative integer \(i\) and a partition \(i=i_{0}+i_{1}+\cdots +i_{n-1}\) into integer vector \(\mathbf{i}=(i_{0},\ldots , i_{n-1})\) we introduce the shorthand notation
These will be the building blocks of our integral formula and we can explicitly calculate this residue using the expansion
Note that all coefficients of this expansion are positive.
Proposition 9.3
-
1.
\(\int _{\mathcal{X}_{n}^{\mathrm{GIT}}} I_{n,\delta }\) is a polynomial in \(d\) of degree \(n+1\) with zero constant term:
$$ \int _{\mathcal{X}_{n}^{\mathrm{GIT}}} I_{n,\delta }=p_{n+1}(n, \delta )d^{n+1}+p_{n}(n,\delta )d^{n}+\cdots +p_{1}(n,\delta )d $$where \(p_{i}(n,\delta )\) is linear in \(\delta \) and polynomial in \(n\) for all \(i\).
-
2.
The leading coefficient is \(p_{n+1}(n,\delta )>C^{\mathbf{0}}\left (1- \frac{\delta n^{3}}{\epsilon}\right )\) and \(C^{\mathbf{0}}=C^{\mathbf{0}}(N,n,\epsilon )>0\) is positive.
Proof
The residue in Proposition 9.1 is by definition the coefficient of \(\frac{1}{zw}\) in the Laurent expansion of the rational expression in \(z\), \(n\), \(d\), \(h\) and \(\delta \) on the contour \(|w|\gg |z|\), that is, in \(z/w\). The result is a polynomial in \(n\), \(d\), \(h\), \(\delta \), and in fact, a relatively easy argument shows that it is a polynomial in \(n\), \(d\), \(\delta \) multiplied by \(h^{n}\) Indeed, setting degree 1 to \(z\), \(w\), \(h\) and 0 to \(n\), \(d\), \(\delta \), the rational expression in the residue has total degree \(n-2\). Therefore the coefficient of \(\frac{1}{zw}\) has degree \(n\), so it has the form \(h^{n} p(n,d,\delta )\) with a polynomial \(p\). Since \(\int _{X} h^{n}=d\), integration over \(X\) is simply a substitution \(h^{n}=d\), resulting in the equation \(\int _{\mathcal{X}_{n}^{\mathrm{GIT}}} I_{n,\delta }=dp(n,\delta ,d)\) for some polynomial \(p(n,\delta ,d)\). The highest power of \(d\) in \(p(n,\delta ,d)\) is \(d^{n}\) which proves the first part.
To prove the second part note that to get \(d^{n+1}\) in Proposition 9.1 we have two options. We drop the \(N^{n^{2}}\) factor from the formulas below.
(i) The first is to choose the \(\frac{dh}{w-z}\) and \(\frac{dh}{lz}\) terms in the product \(\left (1+\frac{dh}{w-z}\right ) \prod _{l=1}^{n}\left (1+ \frac{dh}{lz} \right )\), this contributes with
where \(\Gamma ^{(n+2)}_{j}=\mathrm{coeff}_{(z/w)^{j}} \left (1+\frac{z}{w}+ \frac{z^{2}}{w^{2}} +\cdots \right )^{n+2}\).
(ii) Alternatively we can pick all but one \(dh\) terms from the product \(\left (1+\frac{dh}{w-z}\right ) \prod _{l=1}^{n}\left (1+ \frac{dh}{lz} \right )\) and the \(2\delta n^{3}dh\) term from \(I_{n,\delta }(z,w,h)\). This way the contribution is -\(\sum _{s=0}^{n-1}2 \delta n^{3} C^{\mathbf{e_{s}}}\) where \(\mathbf{e}_{s}\) is the unit vector with all but the \(s\) coordinate zero. From (9.4) these terms are
All coefficients are positive, and by a) comparing the \(i\)th term in \(C^{\mathbf{0}}\) with the \(i-1\)th term in \(C^{\mathbf{e}_{s}}\) when \(1\le s \le n-1\) for \(i\ge 1\) and b) comparing the \(i\)th term in \(C^{\mathbf{0}}\) with the \(i\)h term in \(C^{\mathbf{e}_{s}}\) when \(s=0\), we obtain
holds and hence the total contribution is less than \(2\delta n^{3} C^{\mathbf{0}}(1+\frac{1}{4\epsilon})< \frac{\delta n^{3} C^{\mathbf{0}}}{\epsilon}\) if \(0< \epsilon <1\). Thus the second part is proved. □
We obtain the following corollary.
Proposition 9.4
If \(\delta < \frac{\epsilon}{n^{3}}\) then the leading coefficient \(p_{n+1}(n,\delta )>0\) is positive, and therefore \(\int _{\mathcal{X}_{n}^{\mathrm{GIT}}} I_{n,\delta }>0\) for \(d\gg 0\).
Note that we will choose \(\epsilon =1/2\) after calibrating the parameters.
9.3 Proof of positivity of \(I(n,\delta ,d)\)
According to Proposition 9.3 we have to prove the positivity of the polynomial \(\int _{\mathcal{X}_{n}^{\mathrm{GIT}}}=p_{n+1}(n,\delta )d^{n+1}+p_{n}(n, \delta )d^{n}+\cdots +p_{1}(n,\delta )d\). The strategy is to show that for small enough \(\delta \) the other coefficients satisfy
for \(1\le l \le n+1\). Then we can apply the following elementary statement.
Lemma 9.5
Fujiwara bound
If \(p(d)=p_{n+1}d^{n+1}+p_{n}d^{n}+\cdots +p_{1}d+p_{0}\in {\mathbb{R} }[d]\) satisfies the inequalities
then \(p(d)>0\) for \(d>2D\).
We start with the study of the next coefficient, \(p_{n}\). Similarly to the proof of Proposition 9.3 (2), here we can distinguish four cases how we can get \(d^{n}\) in the residue formula of Proposition 9.1.
(i) If we take \(n-1\) \(dh\) terms from the product \(\left (1+\frac{dh}{w-z}\right ) \prod _{l=1}^{n-1}\left (1+ \frac{dh}{lz} \right )\) and one \(h\) from \(I_{n,\delta }(z,w,h)\) then we get
(ii) If we take \(n-1\) \(dh\) terms from \(\left (1+\frac{dh}{w-z}\right ) \prod _{l=1}^{n-1}\left (1+ \frac{dh}{lz} \right )\) (namely, we drop the \(s\)th \(dh\) term) and one \(h\) from \((1-\frac{h}{w-z}+\cdots )^{n+2}\prod _{l=1}^{n-1}(1-\frac{h}{lz}+ \cdots )^{n+2}\) (namely, the \(h\) in the \(t\)th term) then the contribution is
(iii) If we take \(n-2\) terms from \(\left (1+\frac{dh}{w-z}\right ) \prod _{l=1}^{n-1}\left (1+ \frac{dh}{lz} \right )\), one \(dh\) from \(I_{n,\delta }(z,w,h)\) and one \(h\) from \((1-\frac{h}{w-z}+\cdots )^{n+2}\prod _{l=1}^{n-1}(1-\frac{h}{lz}+ \cdots )^{n+2}\) then the contribution is
(iv) Finally, if we take \(n-2\) terms from \(\left (1+\frac{dh}{w-z}\right ) \prod _{l=1}^{n-1}\left (1+ \frac{dh}{lz} \right )\), one \(dh\) and one \(h\) from \(I_{n,\delta }(z,w,h)\) then the contribution is
Using the positivity of the Taylor expansion, we obtain the following extension of (9.5), with the exact same proof using (9.4).
Lemma 9.6
-
1.
For any integer partition \(\mathbf{i}=(i_{1},\ldots , i_{n-1}\) with \(0\le i_{0}+\cdots +i_{n-1}< n\) we have
$$ C^{\mathbf{i}+\mathbf{e}_{s}}< \textstyle\begin{cases} \frac{C^{\mathbf{i}}}{2n \epsilon} & 1\le s \le n-1 \\ C^{\mathbf{i}} & s=0 \end{cases} $$and hence
$$ \frac{C^{\mathbf{i}}}{n}< \sum _{s=0}^{n-1} C^{\mathbf{i}+ \mathbf{e_{s}}} < C^{\mathbf{i}}\left (1+\frac{1}{\epsilon}\right ) $$ -
2.
If \(i_{0}+\cdots + i_{n-1}=0\) and \(p=1/2(|i_{0}|+\cdots +|i_{n-1}|)\) is the sum of the positive elements of \(\mathbf{i}\) then
$$ C^{\mathbf{i}}< n^{p}C^{\mathbf{0}} $$
Hence
Moreover, if \(\delta =O(\frac{1}{n^{3}})\) holds, then the dominant contributions of \(p_{n}\) are \(B\) and \(C\), and they give
Similar computation shows that the dominant part in \(p_{n-s}\) for \(0\le s \le n\) are the terms corresponding to the choice when we take \(n-s-2\) terms from \(\left (1+\frac{dh}{w-z}\right ) \prod _{l=1}^{n}\left (1+ \frac{dh}{lz} \right )\) one \(dh\) and \(h^{u}\) from \(I_{n,\delta }(z,w,h)\) and \(h^{s+1-u}\) from \((1-\frac{h}{w-z}+\cdots )^{n+2}\prod _{l=1}^{n-1}(1-\frac{h}{lz}+ \cdots )^{n+2}\). This contribution is less than
Note that \(s+1-u=0\) is allowed, in this case the second sum is empty. Applying the second part of part of Lemma 9.6 with \(p=s+1-u\), then the first part \(s+2-(s+1-u)=u+1\) times, we get
Hence the sum (9.7) is less than
and
To finish the proof, we calibrate \(\epsilon \) and \(\delta \) to give the best bound. We first fix \(\epsilon =1/4\), \(\delta =\frac{\epsilon}{4n^{3}}=\frac{1}{16n^{3}}\). With this choice we have:
-
\(p_{n+1}(n,\delta )>C^{\mathbf{0}}\left (1- \frac{\delta n^{3}}{\epsilon}\right )=\frac{3}{4}C^{\mathbf{0}}>0\);
-
\(|p_{n-s}|<\delta 12^{s+2} n^{4s+7}C^{\mathbf{0}}=\frac{3}{4}(2n)^{4s+4}C^{ \mathbf{0}}\).
The Fujiwara estimation of Lemma 9.5 works with \(D=(2n)^{4}\). Hence for \(\delta =\frac{1}{16n^{3}}\) and \(d>2(2n)^{4}\) the integral \(\int _{\mathcal{X}_{n}^{\mathrm{GIT}}} I_{n,\delta }>0\) is positive, hence it provides the existence of nonzero sections in Theorem 6.4. Then Theorem 6.5 applied with
finishes the proof of Theorem 1.3.
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Bérczi, G., Kirwan, F. Non-reductive geometric invariant theory and hyperbolicity. Invent. math. 235, 81–127 (2024). https://doi.org/10.1007/s00222-023-01219-z
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DOI: https://doi.org/10.1007/s00222-023-01219-z