Abstract
An explicit formula for analytic torsion forms for fibrations by projective curves is given. In particular one obtains a formula for direct images in Arakelov geometry in the corresponding setting. The main tool is a new description of Bismut’s equivariant Bott–Chern current in the case of isolated fixed points.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The purpose of this paper is to make the computation of analytic torsion forms more accessible, exploring a method relying on a result by Bismut and Goette. We use this method to explicitly compute analytic torsion forms for fibrations by projective curves.
Analytic torsion forms have been constructed and investigated by Bismut and the author in [14] using heat kernels of certain differential operators. This definition followed previous constructions by Gillet and Soulé [21], Bismut et al. [15], Gillet and Soulé [23]. Further constructions and extensions have been given by Faltings [18], Zha [38], Ma [33], Bismut [12], Burgos Gil-Freixas i Montplet-Liţcanu [16] (including an axiomatic characterisation) and several other articles and books. Analytic torsion forms \(T_\pi (\overline{E})\) are differential forms on the base B associated to Hermitian holomorphic vector bundles \(\overline{E}\) over fibrations \(\pi :M\rightarrow B\) of complex manifolds equipped with a certain Kähler structure. Their degree 0 part equals Ray–Singer’s complex analytic torsion. Their main application is the construction of a direct image \(\pi _!\) of Hermitian vector bundles in Gillet–Soulé’s Arakelov K-theory of arithmetic schemes. This direct image is the sum of higher direct images on algebraic schemes plus \(T_\pi \). Bismut’s immersion formula for torsion forms [9] enabled Gillet–Rössler–Soulé to prove a Grothendieck–Riemann–Roch theorem in Arakelov Geometry which relates \(\pi _!\) to the direct image in Gillet–Soulé’s Chow intersection theory of cycles and Green currents [25], extending [24]. The torsion form also played a key role in Fu’s and Zhang’s proof of the birational invariance of BCOV torsion [20, 39].
While there are many computations for the degree 0 part of analytic torsion forms, there are currently only few explicitly known values of analytic torsion forms in higher degree: torsion forms are known for vector bundles over torus bundles [31]. Also Mourougane showed \(T_\pi (\mathcal {O})^{[2]}=0\) as the value in degree 2 of \(T_\pi \) for the fibration by Hirzebruch surfaces over \(\textbf{P}^1\textbf{C}\) [35]. Furthermore Bismut has shown that the equivariant torsion form of the \(\textbf{Z}\)-graded holomorphic de Rham complex of the fibers vanishes in cohomology [11]. Puchol proved a formula for asymptotic expansion of torsion forms for high powers of a line bundle [36], extended in degree 0 by Finski [19]. In [10, Remark 8.11], the explicit calculation of torsion forms for projective bundles was stated as an open problem with useful applications. We try to improve the situation by proving the following result in Sect. 10:
Theorem 1.1
Let \(\pi :E\rightarrow B\) be a holomorphic vector bundle of rank 2 over a complex manifold. Consider the formal power series \(T_\ell \in \textbf{R}[[X]]\) given by
where \((t^{2m})^*:=t^{2m}\cdot (2\mathcal {H}_{2m+1}-\mathcal {H}_m)\) with the harmonic numbers
The torsion form for \(\overline{\mathcal {O}(\ell )}\) on the \(\textbf{P}^1\textbf{C}\)-bundle \(\pi :\textbf{P}E\rightarrow B\) is given by \( T_\pi (\overline{\mathcal {O}(\ell )})=e^{-\frac{\ell }{2}c_1(E)}T_\ell (c_1(E)^2-4c_2(E))\in H^{2\bullet }(B,\textbf{R}). \)
Our method is as follows. It has been pointed out (after Atiyah and Singer [2, p. 133–134]) by Bismut [5, p. 100–101], Bismut et al. [15] and Berline et al. [4, ch. 10.7] that Bismut super connections and torsion forms can be understood in a more accessible way if the fibration \(\pi :M\rightarrow B\) and the associated objects are induced by a principle bundle \(P\rightarrow B\) with compact structure group G. Bismut and Goette [13] use this to describe the torsion form in such a setting as a cohomology class which can be interpreted in terms of a \(\mathfrak {g}\)-equivariant analytic torsion.
Bismut–Goette’s main result relates this \(\mathfrak {g}\)-equivariant analytic torsion to the G-equivariant analytic torsion introduced in [29] via Bismut’s equivariant Bott–Chern current S. The construction of this Bott–Chern current was inspired by Mathai–Quillen’s very influential construction of a Gauß shape representative of the Thom class [34]. The crucial Gauß density in this construction makes explicit integration in our example difficult, and thus our strategy is to replace it by an indicator function closer to Thom’s original construction (Theorem 5.5). We do this in a general setting for isolated fixed points as this construction shall be applied to more general spaces in a forthcoming paper.
We also employ the formula for S to demonstrate the usage of the residue formula in Arakelov theory [32, Th. 2.11] by applying it to \(\textbf{P}^1_\textbf{Z}\) in Sect. 8. This residue formula (à la Bott) has never been applied before as the S-current makes explicit evaluations difficult. In Theorem 9.3 we give an explicit formula for the \(\mathfrak {g}\)-equivariant analytic torsion on \(\textbf{P}^1\textbf{C}\). In Theorem 12.2, we extend this to \((\mathfrak {g},G)\)-equivariant analytic torsion providing the G-equivariant torsion form introduced in [33]. In Remark 9.5 we verify that the degree 0 part of the formula in Theorem 1.1 equals the known value of the Ray–Singer analytic torsion as given in [30, Theorem 18]. Theorem 11.2 shows that the last summand in Theorem 1.1 exactly cancels with another term in the arithmetic Grothendieck–Riemann–Roch Theorem from [25].
The author is indebted to the referee for a careful reading of this paper and for his comments.
2 Equivariant characteristic classes
Let M be a complex manifold. Corresponding to the decomposition \(TM\otimes \textbf{C}=TM^{1,0}\oplus TM^{0,1}\) define \(U=U^{1,0}+U^{0,1}\) for \(U\in TM\otimes \textbf{C}\). Let \(\mathfrak {A}^{p,q}(M)\) denote the vector space of forms of holomorphic degree p and anti holomorphic degree q, and let \(\mathfrak {A}^{p,q}(M,E)\) denote the corresponding forms with coefficients in a holomorphic vector bundle E. Let \(X\in \Gamma (M,TM)\) be a vector field such that its local flow acts holomorphically on M, i.e. \(X^{1,0}\) is a holomorphic section of \(T^{1,0}M\).
An X-equivariant holomorphic vector bundle E equipped with an X-invariant Hermitian metric shall be denoted by \(\overline{E}\). Let \(\nabla ^E\) be the associated Chern connection with curvature \(\Omega ^E\in \mathfrak {A}^{1,1}(M,\textrm{End}\, E)\).
Following [13, (2.7)] we denote by \(m^E(X):=\nabla ^E_{X}-L^E_{X}\in \Gamma (M,\textrm{End}\,E)\) the moment map as the skew adjoint endomorphism given by the difference between the Lie derivative and the covariant derivative on E. In particular, for the flow \(\Phi _t^X\) associated to X and a zero p of X, \(m^{TX}(X)(p)=\frac{\partial }{\partial t}\big |_{t=0}\Phi ^X_t(p)\in \textrm{End}\, T_pM\). Set as in [13, (2.30), Def. 2.7] (compare [4, ch. 7])
and
The Chern class \(c_{q,X}(\overline{E})\) for \(0\le q\le \textrm{rk}\,E\) is defined in [32, Def. 2.5] as the part of total degree \({\textrm{deg}\,}_M+{\textrm{deg}\,}_t=q\) of
at \(t=1\), thus \(c_{q,X}(\overline{E})=c_q(-\Omega ^E/2\pi i+m^E(X))\). For \(m^E\) invertible we set [13, (3.10)]
The bundle E splits at every component of the fixed point set \(M_X:=\{p\in M\mid X_p=0\}\) into a sum of holomorphic vector bundles \(\bigoplus E_\vartheta \) associated to eigenvalues \(i\vartheta \in i\textbf{R}\) of \(m^E\). Let \(I_{X}\in H^\bullet (M_X)\) denote the additive equivariant characteristic class which is given for a line bundle L as follows: If \(X'\) acts at the fixed point p by an angle \(\vartheta '\in \textbf{R}^\times \) on L, then
Next consider a holomorphic action g on M. Assume that E is g-invariant as a holomorphic Hermitian bundle and that E is equipped with an equivariant structure \(g^E\). The Hermitian vector bundle \(\overline{E}\) splits on the fixed point submanifold \(M_g\) into a direct sum \(\bigoplus _{\zeta \in S^1}\overline{E}_\zeta \), where the equivariant structure \(g^E\) of E acts on \(\overline{E}_\zeta \) as \(\zeta \). Then the g-equivariant Chern character form is defined as
Thus, \({\widetilde{\textrm{ch}}}_g(\overline{E})=\sum _\zeta \zeta {\widetilde{\textrm{ch}}}(\overline{E}_\zeta )\). With the g-invariant subbundle \(E_1\rightarrow M_g\), the Todd form of a g-equivariant vector bundle is defined as
3 Analytic torsion forms
In this section we describe the definition of equivariant Ray–Singer analytic torsions and analytic torsion forms. We simplify the more general setting in [14] a bit for the sake of exposition, as we shall use the torsion forms in this article only in a very restricted setting.
Let M be a compact Kähler manifold of complex dimension \({n}\) with Kähler form \(\omega ^{TM}\in \mathfrak {A}^{1,1}(M)\). We choose the Kähler form such that it verifies the condition \(\omega ^{TM}(U,V)=g^{TM}(JU,V)\) (note that [14, 13, p. 1302] use \(-\omega ^{TM}\) instead as the Kähler form). Thus \(\omega ^{TM}_p=\sum _{j=0}^{{n}}dx_{2j-1}\wedge dx_{2j}\) in geodesic coordinates at an origin p. For \(U\in T^{0,1}M\), \(q\in \textbf{N}_0\), let \(\iota _U:\Lambda ^{q} T^{*0,1}M\rightarrow \Lambda ^{q-1} T^{*0,1}M\) denote the interior product antiderivation. Define fibrewise an action of the Clifford algebra assciated to \((TM,g^{TM})\) on \(\Lambda ^\bullet T^{*0,1}M\otimes E\) by
Let \(N_\infty :\Lambda ^\bullet T^*M\otimes E\rightarrow \textbf{N}_0\) map each component to its differential form degree. Consider a holomorphic isometric action of a Lie group G on M. Consider \(g\in G\) and a vector field X induced by an element of the Lie algebra \(\mathfrak {z}_{G}(g)\subset \mathfrak {g}\) of the centralizer of g. As above let \({\bar{E}}\rightarrow M\) be an X-equivariant Hermitian holomorphic vector bundle.
Assume that the action of X on \((M,\omega )\) is Hamiltonian, i.e. there exists a function \(\mu \in C^\infty (M,\textbf{R})\) such that \(d\mu =\iota _X\omega \). This implies \(X.\mu =0\), and for M connected \(\mu \) is uniquely determined up to constant. If \(L\rightarrow M\) is an X-invariant polarized variety and \(\omega :=i\Omega ^L\), one can choose \(\mu :=-i\cdot m^L\). With the Dolbeault operator associated to E, set as in [13, (2.40)]
acting on \(\mathfrak {A}^{0,\bullet }(M,E)\).
Definition 3.1
[13, p. 1319] For \(s\in \textbf{C}\), \(\textrm{Re}\,s\in ]0,\frac{1}{2}[\) and |X| sufficiently small, the zeta function
is well-defined and Z has a holomorphic continuation to \(s=0\). The (g, X)-equivariant complex Ray-Singer torsion is defined as \(T_{g,X}(M,\overline{E}):=\frac{\partial }{\partial s}\big |_{s=0}Z(s)\).
The g-equivariant torsion \(T_g(M,\overline{E})\) was defined in [29], and Bismut-Goette’s Definition extends this such that \(T_g(M,\overline{E})=T_{g,0}(M,\overline{E})\).
Definition 3.2
[15, Def. 1.4] Let \(\pi :M{\mathop {\rightarrow }\limits ^{Z}}B\) be a proper holomorphic submersion of complex manifolds M, B. Let TZ and \(TZ^\perp \) denote the vertical tangent bundle and the horizontal distribution othogonal to it, respectively. Suppose that there exists a closed 2-form \(\omega \in \mathfrak {A}^{1,1}(M)\) such that \(g^{TZ}:=\omega \big |_{TZ^{\otimes 2}}(\cdot ,J\cdot )\) is Hermitian. Then \((\pi ,g^{TZ},TZ^\perp )\) is called a Kähler fibration.
For \(b\in B\) set \(Z_b:=\pi ^{-1}(\{b\})\). For any \(U\in T_bB\) we denote by \(U^H\in (TZ_b)^\perp \subset TM\) the horizontal lift to the orthogonal complement of the vertical tangent space. Let \(g^{TB}\) be a metric on B, inducing \(\nabla ^{(TZ^\perp )}\). Set \(\nabla ^{TM}:=\nabla ^{(TZ^\perp )}\oplus \nabla ^{TZ}\) with torsion \(\mathcal {T}\in {\mathfrak {A}}^{1,1}(M,TZ)\). Set \(\omega ^H\in {\mathfrak {A}}^{1,1}(B)\otimes C^\infty (M)\), \(\omega ^H(U,U'):=\omega (U^H,{U'}^H)\), \(\mathcal {T}^H(U,U'):=\mathcal {T}(U^H,{U'}^H)\). Let \({\bar{E}}\rightarrow M\) be an Hermitian holomorphic vector bundle. Let \(F\rightarrow B\) denote the \(\infty \)-dimensional vector bundle with fibre
and connection \(\nabla ^F_Us:=\nabla ^{\Lambda ^\bullet T^{*0,1}Z\otimes E}_{U^H}s\).
Definition 3.3
[14, Def. 1.8] For \(t\in \textbf{R}^+\), the number operator \(N_t\in \Gamma (B,\Lambda ^\bullet T^*B\otimes \textrm{End}\, F)\) is given by
The Bismut super connection on F is defined using the Clifford operation c of the TZ component of \(\mathcal {T}\) on \(\Lambda ^\bullet T^{*0,1}Z\) as
The operator \(B_t\) is formed as an adiabatic limit of the Dirac operator on M. As differential forms on the manifold B, the summands have the degrees 1, 0, 2.
Definition 3.4
[14, Def. 3.8] Set \(\tilde{\mathfrak {A}}(B):=\bigoplus _p{\mathfrak {A}^{p,p}(B)}/(\textrm{im}\,\partial +\textrm{im}\,{\bar{\partial }})\). Assume \(H^\bullet (Z_\cdot ,E\big |_{Z_\cdot })\rightarrow B\) to be vector bundles. For \(|\textrm{Re}\,s|<\frac{1}{2}\) set (regularized as in [14, (3.10)])
The analytic torsion form associated to the Kähler fibration \(\pi \) and \({\bar{E}}\) is defined as \( T_\pi ({\bar{E}}):=Z'(0)\in \tilde{\mathfrak {A}}(B). \)
In degree 0 one gets \(T_\pi ({\bar{E}})^{[0]}\big |_{b}=T_\textrm{id}(Z_b,{\bar{E}}\big |_{Z_b})\) with the Ray-Singer torsion on the right hand side.
Example 1
1) [31, Th. 4.1] Let \({\bar{E}}\rightarrow B\) be an Hermitian holomorphic vector bundle of rank k, \(\Lambda \subset E\) a \(\textbf{Z}^{2k}\)-bundle with holomorphic local sections. Then \(\pi :M:=E/\Lambda \rightarrow B\) is a torus bundle. For \(\textrm{Re}\,s<0\) set
Then \(T_\pi (\mathcal {O})=\frac{Z'(0)}{\textrm{Td}({\bar{E}})}\in \tilde{\mathfrak {A}}(B)\). A Kähler fibration condition is not necessary.
2) Mourougane considered Hirzebruch surfaces
and obtained 0 as the part of \(T_\pi (\mathcal {O})\) in degree 2 ( [35, p. 239]).
4 Bismut’s equivariant Bott–Chern current
The equivariant Bott–Chern current has been introduced and investigated by Bismut in [6, 8]. In this section we briefly cite some of its properties following the presentation in [13]. In the special case of isolated fixed points we shall necessarily obtain these results independently in the next section to obtain our expression for \(S_X\). Let M be a compact Kähler manifold acted upon by a holomorphic Killing field X. Denote [13, p. 1312, Def. 2.6]
The holomorphy of X implies \(\partial _X^2=0\), \({\bar{\partial }}_X^2=0\). Notice that \(d_X^2=-2\pi i L_{{X}}\).
Let \(N^*_{\textbf{R}}\) denote the dual of the real normal bundle of the embedding \(M_X\hookrightarrow M\). Let \(P_{X,M_X}^M\) be the set of currents \(\alpha \) on M with wave front set included in \(N^*_{\textbf{R}}\), such that \(\alpha \) is a sum of currents of type (p, p) and \(L_{{X}}\alpha =0\) [13, Def. 3.5]. Let \(P_{X,M_X}^{M,0}\subset P_{X,M_X}^M\) denote the subset consisting of those \(\alpha =\partial _X\beta +{\bar{\partial }}_X\beta '\), where \(\beta ,\beta '\) are \({X}\)-invariant currents whose wave front set is included in \(N^*_{\textbf{R}}\). We shall use the notation \(X^\flat \in \Gamma (M,T^*M)\) for the metric dual of a vector field \(X\in \Gamma (M,TM)\). Set [13, p. 1322, Def. 3.3]
Then [13, (3.9)] there is a current \(\rho _1\in P^M_{X,M_X}\) such that for \(t\rightarrow 0^+\) and any \(\eta \in \mathfrak {A}(M)\),
By Eq. (4), \(F^2_\eta (s):=\frac{1}{\Gamma (s)}\int _1^\infty (\int _M\eta d_t)t^{s-1}dt\) is well-defined and holomorphic for \(\mathrm{Re\,}s<1\) ( [13, (3.13)]). Similarly Eq. (5) shows that
is well-defined for \(\mathrm{Re\,}s>1\) and that it has a holomorphic continuation to \(s=0\). Thus one can set [13, p. 1324, Def. 3.7] \(\int _M\eta S_X(M,-\omega ^{TM}):=\frac{\partial }{\partial s}\big |_{s=0}(F^1_\eta +F^2_\eta )\). By [13, Th. 3.9] \(S_X(M,-\omega ^{TM})\in P^M_{X,M_X}\). By [13, Prop. 3.8] or [8, Prop. 2.11] one gets
According to [13, p. 1323, Th. 3.6] (or [6, (40)–(49)], [8, Th. 2.7])
up to currents in \(P^{M,0}_{X,M_X}\). Thus when \(\eta \) is a \(d_X\)-closed form,
By replacing the first integral by \(\lim _{a\rightarrow 0}\int _a^1\) and using \(\int _a^1\frac{dt}{t^2}+1=\frac{1}{a}\), \(\int _a^1\frac{dt}{t}=-\log a\), \(\int _a^\infty e^{c/t}\frac{dt}{t^2}{\mathop {=}\limits ^{\textrm{Re}\,c<0}}\frac{e^{c/a}-1}{c}\) (and the variant of the last equation for \(\int _a^\infty e^{c/t}\frac{dt}{t^{2+m}}\), \(m\ge 0\)) as in [32, p. 96] this becomes
We shall need in the case of isolated fixed points a sharper version of Eqs. (5), (6): \(\int _M\eta \cdot d_t=\int _{M_X}\eta (c_{\textrm{top},X}^{-1})'(TM)+O(t)\) for any smooth form \(\eta \) (Lemma 5.4) and not only for \(d_X\)-closed forms.
The dependence of \(S_X(M,-\omega ^{TM})\) on \(\omega ^{TM}\) is analysed in [13, Th. 3.10]:
Theorem 4.1
[13, Th. 3.10] For Kähler forms \(\omega ^{'TM},\omega ^{TM}\) on M and the induced metrics \(g^{'TM},g^{TM}\) on \(N_{M_X/M}\),
in \(P^M_{X,M_X}/P^{M,0}_{X,M_X}\).
An important special case arises when rescaling \(\omega ^{'TM}=c^2\omega ^{TM}\): We shall use the notation \(\alpha ^{[q]}\) for the part in degree q of a differential form \(\alpha \). Because of \(\widetilde{\textrm{ch}^{[q]}}(E,c^2h^E,h^E)=\left( {\widetilde{\textrm{ch}}}(\textbf{C},c^2|\cdot |^2,|\cdot |^2)\textrm{ch}(E,h^E)\right) ^{[q-2]}\) by [22, (1.3.5.2)] and \({\widetilde{\textrm{ch}}}(\textbf{C},c^2|\cdot |^2,|\cdot |^2)=-\log c^2\) ( [22, (1.2.5.1)]), one finds
Thus this relation holds when replacing \(\textrm{ch}^{[q]}\) by any other polynomial in the Chern classes, in particular
This way Theorem 4.1 implies a useful formula for the dependence of \(S_{tX}\) on t, which we shall verify independently for isolated fixed points on the level of currents in Corollary 5.6:
Corollary 4.2
Let \(N_\infty :\mathfrak {A}^\bullet (M)\rightarrow \textbf{N}_0\) denote the number operator. Then
in \(P^M_{X,M_X}/P^{M,0}_{X,M_X}\).
Note that \(P^M_{X,M_X},P^{M,0}_{X,M_X}\) are invariant under rescaling of X.
Proof
When replacing \(\omega ^{TM}\) by \(\omega ':=b\omega ^{TM}\) with \(b\in \textbf{R}^+\), the corresponding form \(d_t'\) is given by \(d_t'=d_{t/b}\). On the other hand when replacing X by \({\tilde{X}}:=cX\) with \(c\ne 0\), the associated form \({\tilde{d}}_t\) equals
Thus \(S_{cX}(M,-\omega ^{TM})=c^{-N_\infty /2-1}S_X(M,-c^2\omega ^{TM})\) and the result follows from Theorem 4.1,
\(\square \)
Remark 4.3
If M is compact, its Lie group of isometries is compact and thus the closure of the subgroup generated by a Killing field X is a compact torus T. Thus in this case \(\eta \) can be made X-invariant by taking the mean value \({\tilde{\eta }}\big |_{q}:=\frac{1}{\textrm{vol}T}\int _T (y^*\eta )\big |_{q}\,d\textrm{vol}_y\). As the equivariant Bott–Chern current is X-invariant, one can make the substitution \(\int _M\eta S_X(M,-\omega ^{TM})=\int _M{\tilde{\eta }} S_X(M,-\omega ^{TM})\) and thus always assume that \(L_X\eta =0\). The condition \(d_X\eta =0\) then is a cohomological condition.
5 A formula for the equivariant Bott–Chern current
As in the last section M shall be a compact Kähler manifold and X a holomorphic Killing field. For all of the results in this section, M can as well be any compact subset of a (possibly non-compact) Kähler manifold \({\tilde{M}}\) and X can be a holomorphic Killing field on \({\tilde{M}}\) without any zeros on \(\partial M\), when \(TM,d\textrm{vol}_M,\mathfrak {A}(M)\) are replaced by \(T{\tilde{M}},d\textrm{vol}_{{\tilde{M}}},\mathfrak {A}({\tilde{M}})\).
We assume that the zero set \(M_X=:\{p_\ell \in M\mid \ell \in J\}\) of X has dimension 0. Set
(while for large parts of the results it could be any differential form of degree 2).
Choose R small enough such that the connected component \(B_\ell \) of \(p_\ell \) in \(\{q\in M\mid \Vert X_q\Vert \le R\}\) can be covered by a chart and such that \(B_\ell \) does not contain another zero. For a fixed \(\ell \in J\) we shall denote the corresponding coordinates by x, chosen such that \(x=0\) at \(p_\ell \). Let \(\Vert \cdot \Vert _\textrm{eucl}\), \(d\lambda \) denote the euclidean metric and the Lebesgue measure in this chart.
Part (2) of the following Proposition gives a first estimate for the right hand side in equation (8).
Proposition 5.1
-
1.
There exists \(c_0\in \textbf{R}^+\) such that for all \(s\in \textbf{N}_0\), \(s<{n}\), \(a>0\), \(m\in \textbf{R}\)
$$\begin{aligned} \int _{B_\ell }a^{-m}\Vert X\Vert ^{-2s}e^{-\frac{1}{2a}\Vert X\Vert ^2}d\textrm{vol}_M< & {} c_0a^{{n}-m-s}. \end{aligned}$$ -
2.
For \(a\rightarrow 0^+\) and any \({\tilde{\eta }}\in \mathfrak {A}(M)\) with \({\textrm{deg}\,}{\tilde{\eta }}\ge 2\),
$$\begin{aligned}{} & {} \int _{B_\ell }{\tilde{\eta }}\frac{1-e^{\frac{\nu -\frac{1}{2}\Vert X\Vert ^2}{a}}}{\nu -\frac{1}{2}\Vert X\Vert ^2} -\int _{B_\ell }{\tilde{\eta }}\frac{\nu ^{{n}-1}\left( e^{-\frac{1}{2a}\Vert X\Vert ^2}-1\right) }{\left( \frac{1}{2}\Vert X\Vert ^2\right) ^{{n}}}\\{} & {} \quad = -\sum _{s=1}^{{n}-1} \int _{B_\ell }{\tilde{\eta }}\frac{\nu ^{s-1}}{\left( \frac{1}{2}\Vert X\Vert ^2\right) ^{s}}+ \sum _{m=1}^{{n}-1}\int _{B_\ell }{\tilde{\eta }} e^{-\frac{1}{2a}\Vert X\Vert ^2} \cdot \nu ^{{n}-1}\frac{\left( \frac{1}{2}\Vert X\Vert ^2\right) ^{-{n}+m}}{m!a^{m}} +O(a). \end{aligned}$$
In part (2) the summands on the right hand side converge for \(a\rightarrow 0^+\). In general the summands on the left hand side do not converge.
Proof
1) Choose \(c,C>0\) such that \(c\Vert x\Vert _\textrm{eucl}<\Vert X\Vert \big |_{x}\) and \(d\textrm{vol}_{M|x}=f(x)\cdot d\lambda \) with \(|f|<C\) and, as defined above, \(d\lambda \) being the Lebesgue measure. Then for \(0\le s<{n}\),
Similarly,
The integral on the right hand side exists by inequality (9) (or, of course, classically). Thus there is a constant \(c_0\) depending only on c, C, n and s such that
As s varies over a finite range, \(c_0\) can be chosen independently of s.
2) For \(\frac{1}{2}\Vert X\Vert ^2\ne 0\) Taylor expansion shows
where the sum is finite, as \(\nu \) has vanishing degree 0 part. Additionally expanding \(\exp \) shows
where summands with \(s+m>{n}\) are vanishing. For \(s<{n}\), the integral over the first summand exists by inequality (9). For \(s<{n}\) and \(a\rightarrow 0^+\), by part (1) the integral over the second summand converges if \(s+m\le {n}\) and it equals O(a) for \(s+m<{n}\). \(\square \)
Choose an oriented orthonormal base of \(T_{p_\ell }M\) with
In the corresponding geodesic coordinates,
Proposition 5.2
Define \(a_\ell \) via \(({\tilde{\eta }}\wedge \nu ^{{n}-1})\big |_{p_\ell }=:a_\ell \,d\textrm{vol}_M\). If \({\tilde{\eta }}=\eta \wedge \frac{\omega ^{TM}}{2\pi }\), then
Proof
With \(\omega ^{TM}\big |_{p_\ell }=\sum _{j=1}^{{n}}dx_{2j-1}\wedge dx_{2j}\) we get
\(\square \)
The next Proposition further simplifies the terms in Proposition 5.1(2): (1) computes a limit for the second summand on the right hand side, and (2) simplifies the second summand on the left hand side.
Proposition 5.3
As \(a\rightarrow 0^+\) the following estimates hold:
-
1.
For \({n}=s+m\) and \(0\le s<{n}\)
$$\begin{aligned} \int _{B_\ell }\frac{{\tilde{\eta }}\wedge \nu ^{{n}-1}}{a^mm!\left( \frac{1}{2}\Vert X\Vert ^2\right) ^{s}}e^{-\frac{1}{2a}\Vert X\Vert ^{2}} = a_\ell \frac{\textrm{vol}(S^{2{n}-1})}{\prod ^{{n}}\vartheta _j^2}\frac{2^{{n}-1}}{m}+O(a). \end{aligned}$$ -
2.
For \({n}=s\) and \(B_\ell '(a):=\{x\in B_\ell \mid \Vert X_x\Vert ^2<2a\}\),
$$\begin{aligned}{} & {} \int _{B_\ell } \frac{{\tilde{\eta }}\wedge \nu ^{{n}-1}}{\left( \frac{1}{2}\Vert X\Vert ^2\right) ^{{n}}}\left( e^{-\frac{1}{2a}\Vert X\Vert ^2}-1 \right) +\int _{B_\ell \setminus B_\ell '(a)}{\tilde{\eta }} \frac{\nu ^{{n}-1}}{\left( \frac{1}{2}\Vert X\Vert ^2\right) ^{{n}}}\\{} & {} \quad =\int _{B_\ell '(a)}{\tilde{\eta }} \frac{\nu ^{{n}-1}}{\left( \frac{1}{2}\Vert X\Vert ^2\right) ^{{n}}}\left( e^{-\frac{1}{2a}\Vert X\Vert ^2}-1 \right) +\int _{B_\ell \setminus B_\ell '(a)}{\tilde{\eta }} \frac{\nu ^{{n}-1}}{\left( \frac{1}{2}\Vert X\Vert ^2\right) ^{{n}}} e^{\frac{-\frac{1}{2}\Vert X\Vert ^2}{a}}\\{} & {} \quad = a_\ell \frac{\textrm{vol}(S^{2{n}-1})}{\prod ^{{n}}\vartheta _j^2}2^{{n}-1}\Gamma '(1)+O(a). \end{aligned}$$
We shall use the notation \(B^{2{n}}_R(x)\subset \textbf{R}^{2{n}}\) for the euclidean ball of radius R and center x.
Proof
Replacing the radius R by a smaller number does not affect these statements, as this causes the left hand sides to change by \(O(a^{-m}e^{-\frac{R^2}{2a}})\). Expanding the metric on M in geodesic coordinates at \(p_\ell \) as \(\langle \cdot ,\cdot \rangle \big |_{x}=\langle \cdot ,\cdot \rangle \big |_{0}+\beta \) with \(\beta =O(\Vert x\Vert ^2)\) and \(X_x=Ax\), one gets \(\Vert X_x\Vert ^2=\Vert Ax\Vert _0^2+\beta (Ax,Ax)=\Vert Ax\Vert _0^2+O(\Vert x\Vert ^4)\). As A is invertible, we can choose \(R>0\) sufficiently small such that \(c>0\) exists with \(\left| \Vert X_x\Vert ^2-\Vert Ax\Vert ^2_0\right| <c\Vert Ax\Vert ^4_0\) on \(\{x\in B_\ell \mid \Vert X_x\Vert <R\}\). Then replace R by a smaller value such that \(c\Vert Ax\Vert _0^4<\frac{1}{2}\Vert Ax\Vert _0^2\) on \(\{x\in B_\ell \mid \Vert X_x\Vert <R\}\). By the mean value Theorem \(|y^{-s}-y_0^{-s}|\le |y-y_0|\sup _{|t-y_0|<|y-y_0|}st^{-s-1}\) applied to \(y\mapsto y^{-s}\), one obtains for \(s\in \textbf{R}_0^+\), \(x\ne 0\)
The same way, for \(s\ge 0\)
Applying Proposition 5.1(1) to the right hand side of equation (11) shows that one can replace \(\Vert X\Vert \) by \(\Vert Ax\Vert _0\) in the integrals \(\int _{B_\ell }\frac{{\tilde{\eta }}\wedge \nu ^{{n}-1}}{a^m(\frac{1}{2}\Vert X\Vert ^2)^{s}}e^{-\frac{1}{2a}\Vert X\Vert ^{2}}\) for \(m+s\le {n}\), \(0\le s\le {n}\) up to a term O(a) as \(a\rightarrow 0^+\). Similarly for \(B_\ell ''(a):=\{x\in \textbf{R}^{2{n}}\mid \Vert Ax\Vert ^2_0<2a\}\)
combined with (10) provides this replacement in \(\int _{B_\ell ''(a)}{\tilde{\eta }} \frac{\nu ^{{n}-1}}{(\frac{1}{2}\Vert X\Vert ^2)^{{n}}}\left( e^{-\frac{1}{2a}\Vert X\Vert ^2}-1 \right) \) for the remaining factor-\((-1)\)-term.
Furthermore the integration range \(B_\ell '(a)\) can be replaced by \(B_\ell ''(a)\) for \(m+s={n}\), \(0\le s\le {n}\): One has \(\{x\in B_\ell \mid \Vert Ax\Vert ^2_0+c\Vert Ax\Vert ^4_0<2a\}\subset B_\ell '(a)\subset \{x\in B_\ell \mid \Vert Ax\Vert ^2_0-c\Vert Ax\Vert ^4_0<2a\}\), where \(\Vert Ax\Vert _0^2<\frac{1}{2c}\) on \(B_\ell \). Thus one finds \((B_\ell '(a){\setminus } B_\ell ''(a))\cup (B_\ell ''(a){\setminus } B_\ell '(a))\subset \{x\in B_\ell \mid |\Vert Ax\Vert ^2_0-2a|<c\Vert Ax\Vert ^4_0 \text{ and } \Vert Ax\Vert _0^2<\frac{1}{2c}\}\). The solutions \(A_\pm =\Vert Ax\Vert _0\) of \(\Vert Ax\Vert ^2_0\pm c\Vert Ax\Vert ^4_0=2a\) verify \(|A_\pm -\sqrt{2a}|<c'a^{3/2}\) for a sufficiently small. Then integrals over the difference between \(B_\ell '(a)\), \(B_\ell ''(a)\) are bounded by
Similarly replacing the integration range \(B_\ell \) by \(\{x\in \textbf{R}^{2{n}}\mid \Vert Ax\Vert ^2_0<R^2\}\) causes a difference equal to \(O(a^{-m}e^{-\frac{R^2}{3a}})\).
(1) For \({n}=s+m\), \(m\ge 1\) and \(a\rightarrow 0^+\)
(2) Let Ei denote the exponential integral function given by \(\textrm{Ei}(x)=-\int _{-x}^{+\infty }\frac{e^{-t}}{t}\,dt\) for \(x\in \textbf{R}^-\). For \(s={n}\), \(m=0\) one finds
Using \(|\det A|=\prod ^{{n}}_{j=1}\vartheta _j^2\) the Proposition follows. \(\square \)
Now one can verify as a refinement of Eqs. (5), (6)
Lemma 5.4
For \(t\rightarrow 0^+\) and \(\eta \in \mathfrak {A}(M)\), \(\int _M\eta \cdot d_t=\int _{M_X}\eta (c_{\textrm{top},X}^{-1})'(TM)+O(t).\)
Proof
One finds
\(\square \)
Theorem 5.5
Let X have isolated zeros. For \(a\rightarrow 0^+\), \(B_\ell '(a)=\{x\in B_\ell \mid \Vert X_x\Vert ^2<2a\}\) and \(\eta \in \mathfrak {A}(M)\),
where \(\alpha ^{[<2{n}-2]}\) denotes the part of degree less than \(2{n}-2\) and \(\mathcal {H}_{{n}-1}\) is the harmonic number as in Eq. (1).
Proof
On \(M\setminus \bigcup _{\ell \in J} B_\ell \), \(\left| \int _{M{\setminus }\bigcup _{\ell \in J} B_\ell }\frac{{\tilde{\eta }}}{\nu -\frac{1}{2}\Vert X\Vert ^2}e^{\frac{\nu -\frac{1}{2}\Vert X\Vert ^2}{a}}\right| =o(e^{-\frac{C'}{a}})\) for a constant \(C'>0\) depending on a lower bound for \(\Vert X\Vert ^2\big |_{M\setminus \bigcup _\ell B_\ell }\). On \(B_\ell \) one gets according to Eq. (8) (generalized to this situation by Lemma 5.4) and the previous Propositions
Using the value of \(a_\ell \) as given in Proposition 5.2 finishes the proof. \(\square \)
We check Corollary 4.2 at this point:
Corollary 5.6
Assume that X has isolated zeros. Then
in \(P^M_{X,M_X}/P^{M,0}_{X,M_X}\).
Proof
Consider \(\eta _1\in \bigoplus _q\mathfrak {A}^{q,q}(M)\) and set \(\eta _t:=t^{-N_\infty /2}\eta =\sum _qt^{-q/2}\eta ^{[q]}\) for \(t\in \textbf{R}^+\). By Theorem 5.5,
\(\square \)
When \(d_X\eta _1=0\), then \(d_{tX}\eta _t=0\).
6 The equivariant Bott–Chern current on the projective plane
Consider the line bundle \(\mathcal {L}:=\mathcal {O}(1)\) on the projective line \(M:=\textbf{P}^1\textbf{C}\) with the chart
In these coordinates, the complex structure \(J^{TM}\) is given by \(J^{TM}\frac{\partial }{\partial v}=\cos u\frac{\partial }{\partial u}\). As before, let \(\Omega ^{TM}\) denote the curvature of TM, which in this case equals the curvature tensor of M. For any \(\textbf{SO}(3)\)-invariant metric we find \(-\Omega ^{TM}(\frac{\partial }{\partial v},\frac{\partial }{\partial u})\frac{\partial }{\partial v}=\cos ^2u\cdot \frac{\partial }{\partial u}\). Thus the Fubini-Study form \(\omega ^{TM}:=2\pi c_1(\mathcal {O}(1))=\pi c_1(TM)=\frac{1}{2}J^{TM}\Omega ^{TM}\) is given by \(\omega ^{TM}(\frac{\partial }{\partial v},\frac{\partial }{\partial u})=\frac{\cos u}{2}\). Then
Consider the circle action induced by the vector field \(X:=\frac{\partial }{\partial v}\). Thus, \(\Vert X\Vert ^2=\frac{\cos ^2u}{2}\) and
As \(TM\cong \mathcal {O}(2)\) as \(\textbf{SU}(2)\)-equivariant vector bundles, we find \(m^{\mathcal {O}(1)}=\frac{i}{2}\sin u\) for the corresponding \(\mathfrak {su}(2)\)-action.
Remark 6.1
When instead considering the action of \(\mathfrak {u}(2)\), there is an additional non-trivial constant action of multiples of \(\textrm{id}_{\textbf{R}^2}\) on \(\mathcal {O}(1)\), which induces a constant summand for \(m^{\mathcal {O}(1)}\). We shall not do this in this paper.
For \(\mu :=-im^{\mathcal {O}(1)}\) one finds
as in [13, (2.4)], except that the sign of \(\omega ^{TM}\) is chosen differently.
Let \(\eta \in C^\infty (\textbf{P}^1\textbf{C},\textbf{C})\). Thus in the above coordinates \(\frac{1}{2\pi }\int \eta \big |_{(\begin{array}{c} u\\ v \end{array})}\,dv\) depends smoothly on u and thus on \(\sin u\). Assume that the moment map of X on \(\mathcal {O}(1)\) at the north pole is given by \(m^{\mathcal {O}(1)}=\frac{i}{2}\). Hence it acts by \(m^{N}=i\vartheta =i\) on \(TM=N\) at this point. Thus for the normal bundle \(N\rightarrow \{p\}\) at any fixed point p, one has \((c_{\textrm{top},tX}^{-1})'(\overline{N}) =\frac{-1}{c_{\textrm{top},tX}(\overline{N})}\sum _{\vartheta }\frac{1}{c_1({\bar{N}}_\vartheta )+it\vartheta }=\frac{-1}{(it\vartheta )^2}=\frac{1}{t^2}. \)
Theorem 6.2
For \(\eta \in C^\infty (\textbf{P}^1\textbf{C},\textbf{C})\) set
Then \(\int _{M_X}\eta (c_{\textrm{top},tX}^{-1})'(TM)=\frac{2{\tilde{g}}(1)}{t^2}\) and
Proof
The equivariant Bott–Chern current S is X-invariant. It switches sign under the isometry \(r:(\begin{array}{c} u\\ v \end{array})\rightarrow (\begin{array}{c} -u\\ v \end{array})\), as X is r-invariant and \(r^*\omega ^{TM}=-\omega ^{TM}\). As r changes the orientation, \(\int _M(r^*\eta )S_X(M,-\omega ^{TM})=\int _M\eta S_X(M,-\omega ^{TM})\). Hence in the integrals \(\eta \) and g can be replaced by their mean value \({\tilde{\eta }}\), \({\tilde{g}}\) over the compact orbit of these symmetries. Let \({\tilde{\eta }}_0:={\tilde{g}}(1)\) denote the value of \({\tilde{\eta }}\) at the poles. Applying Theorem 5.5 results in
Using the substitution \(r:=\sin u\) in both integrals on the right hand side, we get the expression in the Theorem as the second integral equals
\(\square \)
The expression in Theorem 6.2 can be given a more combinatorial form for \(\eta \) analytic. For \(m\ge 0\) let \(\mathcal {H}_m=\sum _{j=1}^m\frac{1}{j}\) denote the harmonic numbers as in Eq. (1), in particular \(\mathcal {H}_0=0\).
Theorem 6.3
Set \(\varphi (r)^\#:=\sum _{m=1}^\infty \varphi _m (2\mathcal {H}_{2\,m-1}-\mathcal {H}_{m-1})\) for any complex power series \(\varphi (r)=\sum _{m=0}^\infty \varphi _mr^{2m}\). Assume that \({\tilde{g}}(r)\) is analytic at \(r=0\) with radius of convergence \(>1\). Then
Proof
The integrands Laurent expansion by t in \(\int _{-1}^{1} \left( \frac{2{\tilde{g}}(r)}{t^2} -\frac{2{\tilde{g}}(1)}{t^2} \right) \cdot \frac{dr}{1-r^2}\) provides integrals of the form
for \(m\ge 0\). The result follows by Theorem 6.2. \(\square \)
For the computation of the torsion form we shall need the following variant.
Theorem 6.4
Set \(\varphi (t)^*:=\sum _{m=0}^\infty \varphi _m(2\mathcal {H}_{2\,m+1}-\mathcal {H}_{m})t^{2\,m}\) for any complex Laurent power series \(\varphi (t)=\sum _{m=-1}^\infty \varphi _mt^{2m}\). For \(\eta \in C^\infty (\textbf{P}^1\textbf{C},\textbf{C})\) define g,\({\tilde{g}}\) as in Theorem 6.2 and set \(\eta _{t|(\begin{array}{c} u\\ v \end{array})}:=\eta \big |_{(\begin{array}{c} \arcsin (t\sin u)\\ v \end{array})}\) for \(t\in [-1,1]\).
-
1.
\(\int _{M_X}\eta _t (c_{\textrm{top},tX}^{-1})'(TM)=\frac{2{\tilde{g}}(t)}{t^2}\) and
$$\begin{aligned}{} & {} \int _M\eta _t S_{tX}(M,-\omega ^{TM})\\{} & {} \quad =\int _{-1}^{1} \left( \frac{2{\tilde{g}}(tr)}{t^2} -\frac{2{\tilde{g}}(t)}{t^2} \right) \cdot \frac{dr}{1-r^2} +(\log t^2-2\Gamma '(1))\cdot \frac{2{\tilde{g}}(t)}{t^2}. \end{aligned}$$ -
2.
Assume that \({\tilde{g}}(t)\) is analytic at \(t=0\) with radius of convergence \(>1\). Then
$$\begin{aligned}{} & {} \int _M\eta _t S_{tX}(M,-\omega ^{TM})= -\left( \frac{2{\tilde{g}}(t)}{t^2}\right) ^* +(\log t^2-2\Gamma '(1))\cdot \frac{2{\tilde{g}}(t)}{t^2}\\{} & {} \quad =-\left( \int _{M_X}\eta _t (c_{\textrm{top},tX}^{-1})'(TM)\right) ^*+(\log t^2-2\Gamma '(1))\int _{M_X}\eta _t (c_{\textrm{top},tX}^{-1})'(TM). \end{aligned}$$
Proof
This follows immediately from Theorems 6.2, 6.3 by replacing \(\eta \) with \(\eta _t\) and using \(\left( \frac{{\tilde{g}}(t)}{t^2}\right) ^*=\left( \frac{{\tilde{g}}(tr)}{t^2}\right) ^\#\). \(\square \)
7 The defining property of \(S_X\)
The equivariant Bott–Chern current verifies the critical relation
Theorem 7.1
[13, Th. 3.9] Using the inverse of the equivariant top Chern class, the identity
holds.
In this section we shall quickly illustrate how this relation can be seen using the formula in Theorem 6.2. Because of
when setting \(\eta =:f_1\omega ^{TM}+f_0\) with \(f_0,f_1\in C^\infty (M)\), Theorem 7.1 translates to
In the coordinates u, v as above, \(X^{1,0}=\frac{1}{2}(\frac{\partial }{\partial v}-i\cos u\frac{\partial }{\partial u})\) and \(2\pi i\omega ^{TM}(X^{0,1},X^{1,0})=\frac{\pi }{2}\cos ^2u\). Assume w.l.o.g. that \(f_1\) and \(X^{1,0}.f_0\) are invariant under X and under \((\begin{array}{c} u\\ v \end{array})\mapsto (\begin{array}{c} -u\\ v \end{array})\). The X-invariance of \(X^{1,0}.f_0\) is in fact equivalent to the X-invariance of \(f_0\), as the equation \(\frac{\partial ^2}{\partial v^2} f_0=0\) for the real part implies \(\frac{\partial }{\partial v} f_0=\)const. And \(f_0\) is periodic in v, thus \(\frac{\partial }{\partial v} f_0=0\). Now with
as in Theorem 6.2, Eq. (12) is equivalent to
The real part of \(X^{1,0}.f_0=\frac{1}{2}\frac{\partial f_0}{\partial v}-\frac{i}{2}\cos u\cdot \frac{\partial f_0}{\partial u}\) does not contribute because of the X-invariance of \(X^{1,0}.f_0\). Setting as in Theorem 6.2
one finds
Using \(c_{\textrm{top},X}^{-1}(N_{M_X/M})\big |_{N}=\frac{1}{i}=-i\), \(c_{\textrm{top},X}^{-1}(N_{M_X/M})\big |_{S}=-\frac{1}{i}=i\), Eq. (13) and thus the equation in Theorem 7.1 follows.
8 The height of \(\textbf{P}^1_\textbf{Z}\)
One of the applications of Bismut’s equivariant Bott–Chern current is a residue formula (in the spirit of Bott’s formula) in Arakelov geometry [32]. In this section we verify that the formula gives the correct classically well-known value for the height of the projective plane over \(\textrm{Spec}\,\textbf{Z}\). We refer to [37] for the concepts of Arakelov geometry and for the associated notations. By [37, p. 70], the height of the projective plane \(f:\textbf{P}^1_\textbf{Z}\rightarrow \textrm{Spec}\,\textbf{Z}\) with respect to the line bundle \(\mathcal {L}:=\mathcal {O}(1)\) is given by \( \widehat{{\textrm{deg}\,}}(f_*{\hat{c}}_1(\overline{\mathcal {O}(1)})^2)\in \textbf{R}\) in terms of the Arakelov characteristic class \({\hat{c}}_1\) having values in the Gillet-Soulé intersection theory \(\widehat{CH}(\textbf{P}^1_\textbf{Z})\). Let \(\mathcal {T}:=\textrm{Spec}\,\textbf{Z}[X,X^{-1}]\) be the one-dimensional torus group scheme and consider its canonical action on \(\textbf{P}^1_\textbf{Z}\) with fixed point scheme consisting of two copies of \(\textrm{Spec}\,\textbf{Z}\). Let r denote the additive characteristic class which is defined in [32, p. 90] as
for L a line bundle acted upon by X with an angle \(\varphi \in \textbf{R}\) at \(p\in M_{X}\). According to the residue formula in Arakelov geometry proven in [32, Th. 2.11], the height can be computed using equivariant Arakelov characteristic classes \({\hat{c}}_{1,t}\), \({\hat{c}}_{\textrm{top},t}\) and the normal bundle \({\bar{N}}\) as
Classically, at the fixed point subscheme \(\widehat{c}_1(\overline{\mathcal {L}})=\widehat{c}_1(\overline{N})/2=0\), thus the arithmetic term on the right hand side of the residue formula (14) vanishes. At a fixed point p let tX act by an angle \(\varphi \) on \(\mathcal {O}(1)\) and by an angle \(\vartheta \) on N. In our case the angles \(\varphi \) and \(\vartheta \) at the fixed points are given by \(\pm \frac{t}{2}\) and \(\pm t\), respectively. As in [32, p. 98],
According to Theorem 6.4 we get with \(\eta _t=(m^\mathcal {L}(tX))^2=-\frac{t^2}{4}\sin ^2 u\) and \({\tilde{g}}(t)=-\frac{1}{4}t^2\)
Hence the residue formula in Arakelov geometry [32, Th. 2.11] states in this case
which is the well-known classical value [37, p. 71].
9 Lie algebra equivariant torsion on \(\textbf{P}^1\textbf{C}\)
We employ the following special case of Bismut–Goette’s main result:
Theorem 9.1
[13, Th. 0.1] For |t| sufficiently small,
With the angle \(t\vartheta \) of the operation of \(g=e^{tX}\) on \(T_pM\) at the fixed point p, \(t\ne 0\), we get
By [7, (20) (appendix)], for \(0<|t\vartheta |<2\pi \) the last term \(I_{tX}(\textbf{P}^1\textbf{C})\) equals
using the harmonic numbers as given in Eq. (1). For \(M=\textbf{P}^1\textbf{C}\), \(T\textbf{P}^1\textbf{C}=\mathcal {O}(2)\), \(\vartheta ^{\mathcal {O}(1)}=\pm 1/2\) at the fixed points we get for the second summand on the right hand side in Theorem 9.1
Proposition 9.2
For \(M=\textbf{P}^1\textbf{C}\), \(t\ne 0\), the first summand on the right hand side in Theorem 9.1 is given by
Proof
Setting \(m^{\mathcal {L}}:=m^{\mathcal {O}(1)}\) the X-equivariant classes are given by
Thus we get
and henceforth
Remember that by its definition via the integral \(\int _M\eta \wedge d_t\), where \(d_t\) as in Eq. (4) is a form of degree 2 and higher, in this complex-1-dimensional case \(\int \eta S_{tX}(M,-\omega ^{TM})\) only depends on \(\eta ^{[0]}\) for any form \(\eta \). Hence the result follows by applying Theorem 6.4 with \({\tilde{g}}(t)=\frac{t \cos (\frac{(\ell +1)t}{2})}{2\sin (\frac{t}{2})}\). \(\square \)
Theorem 9.3
For \(0<t<2\pi \), the value in Proposition 9.2 has for \(\ell \rightarrow +\infty \) an asymptotic expansion given by
Note that \(\log (\ell +1)=\log \ell +O(\frac{1}{\ell })\).
Proof
We decompose the integral in Proposition 9.2 as
For \(|t|<2\pi \), the factor \(f(r):=\left( \frac{r}{\sin \frac{tr}{2}}-\frac{1}{\sin \frac{t}{2}} \right) \cdot \frac{1}{t(1-r^2)}\) in the second integral on the right hand side is smooth on \(r\in [-1,1]\). Partial integration shows
To estimate the first integral on the right hand side, we represent it as twice the integral over [0, 1], decompose \(\frac{1}{1-r^2}=\frac{1/2}{1-r}+\frac{1/2}{1+r}\) and use the trigonometric addition formula for \(\cos \frac{(\ell +1)tr}{2}=\cos \left( \frac{(\ell +1)t(r-1)}{2}+\frac{(\ell +1)t}{2}\right) \). Let \(\textrm{Si}\) and \(\textrm{Ci}\) denote the sine and cosine integral functions, respectively, which are given by \(\textrm{Si}(x)=\int _0^x\frac{\sin t}{t}\,dt\) and \(\textrm{Ci}(x)=-\int _x^{+\infty }\frac{\cos t}{t}\,dt\) for \(x\in \textbf{R}^+\). Then the above integral equals
Adding the term \((\log t^2-2\Gamma '(1))\cdot \frac{\cos \frac{(\ell +1)t}{2}}{t\sin \frac{t}{2}}\) from Proposition 9.2, one obtains the result. \(\square \)
By iterating the partial integration, one can extend this expansion to arbitrary negative powers of \(\ell +1\).
Theorem 9.4
With respect to the action of the vector field \(X\in \Gamma (\textbf{P}^1\textbf{C},T\textbf{P}^1\textbf{C})\), the X-equivariant torsion is given by
where \((t^{2\,m})^*:=t^{2\,m}\cdot \left\{ \begin{array}{c} 2\mathcal {H}_{2\,m+1}-\mathcal {H}_{m}\\ 0 \end{array} \text{ if } \begin{array}{c} m\ge 0\\ m=-1 \end{array} \right. \) (as in Theorem 6.4) and \(\mathcal {H}_{m}\) is the harmonic number as in Eq. (1).
The first summand contains exactly the function defining the (non-equivariant) Gillet–Soulé R-class [37, p. 160],
(see Theorem 11.2 for a closer analysis). The \(\zeta '\)-term as well as the equivariant-metric-terms are derived from the equivariant torsion. The \(*\)-summand originates from the equivariant Bott–Chern current and the \(\mathcal {H}_m\zeta (-m)\)-term is the I-class. Some terms from the first two summands cancel each other.
Proof
[29, Th. 2] shows for \(t\in ]0,2\pi [\),
where according to [29, Prop. 1],
Combining this with Bismut–Goette’s Theorem 9.1, Proposition 9.2 and Eq. (9.2) we find
\(\square \)
This sum does not contain a factor \(\log t\) nor any negative powers of t anymore. It is an even power series in t. The expansion in t up to \(O(t^4)\) is given by
Remark 9.5
We shall verify that the value of \(T_{\textrm{id},tX}(\textbf{P}^1\textbf{C},\overline{\mathcal {O}(\ell )})\) for \(t=0\) equals the known formula for \(T(\textbf{P}^1\textbf{C},\overline{\mathcal {O}(\ell )})\):
The equivariant torsion has been computed in [30, Theorem 18] for equivariant vector bundles on symmetric spaces. We shall use the notations \({\varvec{\zeta }},{\varvec{\zeta '}},\chi ^*\) etc. from [30, p. 102]: For \(\varphi \in \textbf{R}\) and \(\textrm{Re}\,s>1\), consider the Lerch zeta function
For \(\varphi \) fixed, the function \(\zeta _L\) has analytic continuation in the variable s to \(\textbf{C}{\setminus }\{1\}\). Set \(\zeta '_L(s,\varphi ):=\partial /\partial s (\zeta _L(s,\varphi ))\). Let \(P:\textbf{Z}\rightarrow \textbf{C}\) be a function of the form
with \(m\in \textbf{N}_0\), \(n_j\in \textbf{N}_0\), \(c_j\in \textbf{C}\), \(\varphi _j\in \textbf{R}\) for all j. Then for \(p\in \textbf{R}\) we shall use the notations \(P^\textrm{odd}(k):=(P(k)-P(-k))/2\),
and
For \(\textbf{P}^1\textbf{C}=\textbf{U}(2)/\textbf{U}(1)\times \textbf{U}(1)=:G/K\) identify the Lie algebra of the maximal torus with \(\textbf{R}^2\) with the ordering \((e_1,e_2)\). Then the positive roots are given by \(\Delta ^+=\psi =\{e_1-e_2\}\), and the weight providing \(\mathcal {O}(\ell )\) is given by \(\Lambda =-\ell \cdot e_2\). Thus for \(\alpha =e_1-e_2\), \(\rho _G=\frac{\alpha }{2}\) we get the dimension \(\chi _{\rho _G+\Lambda +k\alpha }(0)=\frac{\langle \alpha ,\rho _G+\ell \cdot e_1+k\alpha \rangle }{\langle \alpha ,\rho _G\rangle }=1+\ell +2k\). Furthermore \((\alpha ,\rho _G+\Lambda )=\frac{2\langle \alpha ,\rho _G+\ell \cdot e_1\rangle }{\langle \alpha ,\alpha \rangle }=1+\ell .\) The formula in [30, Theorem 18] requires the highest weight \(\Lambda \) of the bundle to be in the closure of the positive Weyl chamber. Thus for \(\ell \ge 0\), one obtains for the evaluation of the characters at the neutral element
for the choice \(\frac{\Vert \alpha \Vert ^2_\diamond }{2}=1\) as in [30, (71)]. Köhler [30, Theorem 18] contained a mistyped sign in the third summand. A formula valid for arbitrary equivariant bundles (and without the typo) was given in [27, Th. 5.2], written slightly differently using \({\varvec{\zeta }}P(k)=-{\varvec{\zeta }}P(-k)-P(0)\). This result shows for any \(\ell \in \textbf{Z}\) that \( T(\textbf{P}^1\textbf{C},\overline{\mathcal {O}(\ell )})=4\zeta '(-1)-\frac{(\ell +1)^2}{2} -\sum _{k=1}^{|\ell +1|}(|1+\ell |-2k)\log k\). For arbitrary \(X_0\) one gets an additional summand
where \(\frac{2}{\Vert \alpha \Vert _\diamond }=\frac{1}{\langle \alpha ,\rho _G\rangle _\diamond }=\textrm{vol}_\diamond \textbf{P}^1\textbf{C}\) by [4, Cor. 7.27]. See also [28, p. 840] for additional remarks.
10 The torsion form
Let \(P\rightarrow B\) be a \(\textbf{U}(2)\) principal bundle, \(\textbf{P}\rightarrow B\) the induced \(\textbf{P}^1\textbf{C}\)-bundle and \(E:=P\times _{\textbf{U}(2)}\textbf{C}^2\). Then \(\textbf{P}=\textbf{P}(E)\). The curvature form \(\Theta \in \Lambda ^{1,1}T^*B\otimes (P\times _{\textbf{U}(2)}{\mathfrak {u}}(2))\) inserted in the torsion form as a \(\textbf{C}\)-valued homogeneous polynomial on \(\mathfrak {u}(2)\) provides an expression in terms of \(c_1(E), c_2(E)\) via the fiber bundle embedding \(P\times _{\textbf{U}(2)}{\mathfrak {u}}(2)\hookrightarrow P\times _{\textbf{U}(2)}\textrm{End}(\textbf{C}^2)=\textrm{End}(E)\). In general [13, (2.74)] shows for such bundles induced by principal bundles with compact structure group that the torsion form is a cohomology class.
Remark 10.1
In general \(\textbf{P}^1\textbf{C}\)-bundles can look more complicated; the relevant structure group is \(\textbf{PU}(2)=\textbf{SO}(3)\) and the obstruction is an element of \(H^3(B,\textbf{Z})\): The structure group of projective bundles is \(\textbf{PU}(k)=\textbf{U}(k)/\textbf{U}(1)=\textbf{SU}(k)/(\textbf{Z}/k\textbf{Z})\) (embedded diagonally). Thus one gets an obstruction \(\alpha \in H^3(B,\textbf{Z})\) with \(k\alpha =0\) (see [26, p. 517]; [1], and [3]). See also [17] for a more detailed discussion of the holomorphic situation.
Proof (of Theorem 1.1)
Each \(Y\in {\mathfrak {u}}(2)\) induces a vector field on \(\textbf{P}^1\textbf{C}=\textbf{U}(2)/\textbf{U}(1)\times \textbf{U}(1)\), which we shall denote by \(\rho (Y)\). The Lie algebra element \(\left( \begin{array}{c} i/2\\ 0 \end{array}\begin{array}{c} 0\\ -i/2 \end{array}\right) \in \mathfrak {u}(2)\) acts with period \(\pi \) on \(\textbf{P}^1\textbf{C}\) and induces the vector field \(X=\frac{\partial }{\partial v}\). Thus the element \(Y_0=\left( \begin{array}{c} i\alpha \\ 0 \end{array}\begin{array}{c} 0\\ i\beta \end{array}\right) \in \mathfrak {u}(2)\) induces the vector field \((\alpha -\beta )\cdot X\). For any \(Y\in \mathfrak {g}\), \(\gamma \in G\) the equality \(T_{\textrm{id},\rho ({\textrm{Ad}_\gamma Y})}(\textbf{P}^1\textbf{C},\overline{\mathcal {O}(\ell )})=T_{\textrm{id},\rho (Y)}(\textbf{P}^1\textbf{C},\overline{\mathcal {O}(\ell )})\) holds, as both vector bundle and metric are \(\gamma \)-invariant. Thus \(T_{\textrm{id},tX}(\textbf{P}^1\textbf{C},\overline{\mathcal {O}(\ell )})\) determines \(T_{\textrm{id},\rho (Y)}(\textbf{P}^1\textbf{C},\overline{\mathcal {O}(\ell )})\) completely. Because of the \(\textrm{Ad}\)-invariance of \(\textrm{Tr}\,,\det \) and \((\textrm{Tr}\,Y_0)^2-4\det Y_0=-(\alpha -\beta )^2\),
Considering the determinant of the Euler sequence for the map \(\pi :\textbf{P}^1\textbf{C}\rightarrow \)point
one finds \(\mathcal {O}(-2)\cong \pi ^*\Lambda ^2\textbf{C}^2\otimes T^*\textbf{P}^1\textbf{C}\). As \(e^{Y_0}\in \textbf{U}(2)\) acts with weight \(e^{i(\alpha +\beta )}\) on the pointwise trivial line bundle \(\pi ^*\Lambda ^2\textbf{C}^2\), the action of \(e^{Y_0}\) on \(\mathcal {O}(\ell )\) is given by the action of the traceless component in \(\mathfrak {su}(2)\) composed with the pointwise factor \(e^{-i\ell \frac{\alpha +\beta }{2}}\). Thus, when considering the torsion with respect to the action of the Lie algebra element \(Y\in \mathfrak {g}\) instead of the action of the vector field \(\rho (Y)\), one gets the value (24) multiplied by \(e^{-\frac{\ell }{2}\textrm{Tr}\,Y}\).
According to [13, (2.74)], one obtains the torsion form \(T_\pi (\overline{\mathcal {O}(\ell )})\) by replacing \(Y\in \mathfrak {g}\) with \(-\frac{1}{2\pi i}\Omega ^E\). Thus in the value of \(T_{\textrm{id},tX}(\textbf{P}^1\textbf{C},\overline{\mathcal {O}(\ell )})\) given by Theorem 9.4, \(-t^2\) has to be replaced by \(c_1(E)^2-4c_2(E)\), and the factor \(e^{-\frac{\ell }{2}\textrm{Tr}\,Y}\) gets replaced by \(e^{-\frac{\ell }{2}c_1(E)}\). \(\square \)
One can also verify quickly that the class \(c_1(E)^2-4c_2(E)\) is invariant under \(E\mapsto E\otimes \mathcal {L}'\) for every line bundle \(\mathcal {L}'\). This verifies that it is indeed well-defined for \(\textbf{P}^1\textbf{C}\)-bundles.
11 Comparison with the arithmetic Grothendieck–Riemann–Roch theorem
Given a \(\textbf{P}^1\textbf{C}\)-bundle \(\pi :\textbf{P}\rightarrow B\), we denote the vertical tangent space by \(T\pi \).
Proposition 11.1
For any \(\textbf{P}^1\textbf{C}\)-bundle \(\pi :\textbf{P}\rightarrow B\) one obtains
Thus \(\pi ^*T_\pi (\overline{\mathcal {O}(\ell )})=T_\ell (c_1(T\pi )^2)\).
Proof
Using the Euler sequence for projective fibrations
one finds
and thus \(\pi ^*c_1(E)=c_1(T\pi )-2c_1(\mathcal {O}(1))\) and
\(\square \)
For a general \(\textbf{P}^1\textbf{C}\)-bundle one has to replace E by \(H^0(\textbf{P}^1\textbf{C},\mathcal {O}(1))\) in the result. The arithmetic Grothendieck–Riemann–Roch theorem [25] states with the fibres \(Z\cong \textbf{P}^1\textbf{C}\) of the fibration \(\textbf{P}E\rightarrow B\)
Theorem 11.2
When multiplied by \(e^{-\frac{\ell }{2}c_1(E)}\), the summand
of \(T_\ell (-t^2)\) contributes the term \(\int _{Z}\textrm{ch}(\mathcal {O}(\ell ))\textrm{Td}(T\pi )R(T\pi )\) to the torsion form.
Proof
By the projection formula \(\pi _*(\pi ^*\alpha \wedge \beta )=\alpha \wedge \pi _*\beta \) in cohomology and Proposition 11.1,
Similarly, one finds
Noticing \({\tilde{T}}_\ell (-(it)^2)=\frac{\cosh \frac{(\ell +1)t}{2}}{\sinh \frac{t}{2}}R(t)\), one gets
\(\square \)
12 Equivariant torsion forms
Consider a holomorphic isometric action of a Lie group G on M. Consider \(g\in G\) and a vector field X induced by an element of the Lie algebra \(\mathfrak {z}_{G}(g)\subset \mathfrak {g}\) of the centralizer of g. Let \(I_{g,X}\) denote the additive equivariant characteristic class on \(M_g\cap M_X\) which is given for a line bundle L as follows: If X acts at the fixed point p by an angle \(\vartheta '\in \textbf{R}\) on L and g acts by \(e^{i\vartheta }\) with \(\vartheta \in [0,2\pi [\), then for \(|\vartheta '|\) sufficiently small
Set \({\tilde{R}}_0(\vartheta ,x):=\sum _{k=0}^\infty \left( \frac{\partial }{\partial s}\zeta _L(-k,\vartheta )+\zeta _L(-k,\vartheta )\frac{\mathcal {H}_k}{2}\right) \frac{x^k}{k!}\) with \(\zeta _L\) as in Eq. (21). For \(\vartheta \ne 0\), set \(R(\vartheta ,x):={\tilde{R}}_0(e^{i\vartheta },x)-{\tilde{R}}_0(e^{-i\vartheta },-x)\). In [29, Prop. 1], it is shown that \(2iR^\textrm{rot}(\vartheta )=R(\vartheta ,0)\) for \(\vartheta \in ]0,2\pi [\) with \(R^\textrm{rot}\) given by Eq. (19). Bismut–Goette showed in equation [13, (0.13)] that for \(\vartheta \ne 0\), \(|\vartheta '|\) sufficiently small,
We refer to [13, Th. 2.7] for the definition of (g, tX)-equivariant characteristic classes \(\textrm{Td}_{g,tX}\), \(\textrm{ch}_{g,tX}\) and torsion \(T_{g,tX}\). Bismut–Goette’s main result shows for the action of \(g\in \textbf{SU}(2)\) and the infinitesimal action of \(X\in \mathfrak {su}(2)\) on \(\textbf{P}^1\textbf{C}\):
Theorem 12.1
[13, Th. 1] Let \(g\in \textbf{SU}(2)\), \(X\in \mathfrak {z}_{\textbf{SU}(2)}(g)\) act of \(\textbf{P}^1\textbf{C}\). Then the (g, tX)-equivariant torsion \(T_{g,tX}(\textbf{P}^1\textbf{C},\overline{\mathcal {O}(\ell )})\) verifies for |t| sufficiently small
If \(g=:e^{sX}\) acts with isolated fixed points, the S-current term disappears, as the S-current has no degree 0 part. Thus
Similarly to Eq. (16), one finds
Thus using Eqs. (18) and [13, (0.13)] one obtains
Theorem 12.2
The \((e^{sX},tX)\)-equivariant torsion verifies for |s|, |t| sufficiently small
This provides the value of the G-equivariant torsion form introduced in [33] analogous to Theorem 1.1.
References
Atiyah, M.F., Anderson, D.W.: K-theory. Mathematics Lecture Notes, vol. 7. W. A. Benjamin, New York (1967)
Atiyah, M.F., Singer, I.M.: The index of elliptic operators. IV. Ann. Math. 93, 119–138 (1971)
Atiyah, M.: K-theory Past and Present. Sitzungsberichte der Berliner Mathematischen Gesellschaft, pp. 411–417. Berliner Math. Ges., Berlin (2001)
Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Ooperators. Grundlehren der Mathematischen Wissenschaften, vol. 298. Springer, Berlin (1992)
Bismut, J.-M.: The Atiyah–Singer index theorem for families of Dirac operators: two heat equation proofs. Invent. Math. 83(1), 91–151 (1986)
Bismut, J.-M.: Equivariant Bott–Chern currents and the Ray–Singer analytic torsion. Math. Ann. 287(3), 495–507 (1990)
Bismut, J.-M.: Koszul complexes, harmonic oscillators, and the Todd class. With an appendix by the author and C. Soulé. J. Am. Math. Soc. 3, 159–256 (1990)
Bismut, J.-M.: Complex equivariant intersection, excess normal bundles and Bott–Chern currents. Commun. Math. Phys. 148(1), 1–55 (1992)
Bismut, J.-M.: Holomorphic families of immersions and higher analytic torsion forms. Astérisque 244, viii+275 (1997)
Bismut, J.-M.: Quillen metrics and singular fibres in arbitrary relative dimension. J. Algebraic Geom. 6(1), 19–149 (1997)
Bismut, J.-M.: Holomorphic and de Rham torsion. Compos. Math. 140(5), 1302–1356 (2004)
Bismut, J.-M.: Hypoelliptic Laplacian and Bott–Chern Cohomology. A Theorem of Riemann–Roch–Grothendieck in Complex Geometry. Prog Math, vol. 305. Birkhäuser/Springer, Cham (2013)
Bismut, J.-M., Goette, S.: Holomorphic equivariant analytic torsions. Geom. Funct. Anal. 10(6), 1289–1422 (2000)
Bismut, J.-M., Köhler, K.: Higher analytic torsion forms for direct images and anomaly formulas. J. Algebraic Geom. 1, 647–684 (1992)
Bismut, J.-M., Gillet, H., Soulé, C.: Analytic torsion and holomorphic determinant bundles. II. Direct images and Bott–Chern forms. Commun. Math. Phys. 115(1), 79–126 (1988)
Burgos Gil, J.I., Freixas i Montplet, G., Liţcanu, R.: Generalized holomorphic analytic torsion. J. Eur. Math. Soc. 16(3), 463–535 (2014)
Coelho, R., Kotschick, D.: Lifts of projective bundles and applications to string manifolds. Bull. Lond. Math. Soc. 53, 470–481 (2021)
Faltings, G.: Lectures on the Arithmetic Riemann–Roch Theorem. Notes Taken by Shouwu Zhang. Annals of Mathematics Studies, vol. 127. Princeton University Press, Princeton (1992)
Finski, S.: On the full asymptotics of analytic torsion. J. Funct. Anal. 275(12), 3457–3503 (2018)
Fu, L., Zhang, Y.: Motivic integration and the birational invariance of the BCOV invariant. Sel. Math. New Ser. 29, 25 (2023)
Gillet, H., Soulé, C.: Direct images of Hermitian holomorphic bundles. Bull. Am. Math. Soc. 15(2), 209–212 (1986)
Gillet, H., Soulé, C.: Characteristic classes for algebraic vector bundles with hermitian metric. I. Ann. Math. 131, 163–203 (1990)
Gillet, H., Soulé, C.: Analytic torsion and the arithmetic Todd genus. With an appendix by D. Zagier. Topology 30(1), 21–54 (1991)
Gillet, H., Soulé, C.: An arithmetic Riemann–Roch theorem. Invent. Math. 110(3), 473–543 (1992)
Gillet, H., Rössler, D., Soulé, C.: An arithmetic Riemann–Roch theorem in higher degrees. Ann. Inst. Fourier 58(6), 2169–2189 (2008)
Ivancevic, V.G., Ivancevic, T.T.: Applied Differential Geometry. A Modern Introduction. World Scientific Publishing Co. Pte. Ltd, Hackensack (2007)
Kaiser, C., Köhler, K.: A fixed point formula of Lefschetz type in Arakelov geometry. III. Representations of Chevalley schemes and heights of flag varieties. Invent. Math. 147(3), 633–669 (2002)
Klevtsov, S., Ma, X., Marinescu, G., Wiegmann, P.: Quantum hall effect and Quillen metric. Commun. Math. Phys. 349(3), 819–855 (2017)
Köhler, K.: Equivariant analytic torsion on \({ P}^{n}{ C}\). Math. Ann. 297(3), 553–565 (1993)
Köhler, K.: Holomorphic torsion on Hermitian symmetric spaces. J. Reine Angew. Math. 460, 93–116 (1995)
Köhler, K.: Complex analytic torsion forms for torus fibrations and moduli spaces. In: Schappacher, N., Reznikov, A. (eds.) Regulators in Analysis, Geometry and Number Theory, Progress in Mathematics, vol. 171, pp. 166–195. Birkhäuser, Basel (2000)
Köhler, K., Roessler, D.: A fixed point formula of Lefschetz type in Arakelov geometry. II. A residue formula. Ann. Inst. Fourier (Grenoble) 52(1), 81–103 (2002)
Ma, X.: Submersions and equivariant Quillen metrics. Ann. Inst. Fourier 50(5), 1539–1588 (2000)
Mathai, V., Quillen, D.: Superconnections, Thom classes, and equivariant differential forms. Topology 25(1), 85–110 (1986)
Mourougane, C.: Analytic torsion of Hirzebruch surfaces. Math. Ann. 335(1), 221–247 (2006)
Puchol, M.: The asymptotic of the holomorphic analytic torsion forms. J. Lond. Math. Soc. 108, 80–140 (2023)
Soulé, C., Abramovich, D., Burnol, J.F., Kramer, J.: Lectures on Arakelov Geometry (Cambridge Studies in Mathematics), vol. 33. Cambridge University Press, Cambridge (1992)
Zha, Y.: A general arithmetic Riemann-Roch theorem. Ph.D. thesis, Univ. of Chicago (1998)
Zhang, Y.: BCOV invariant and blow-up. Compos. Math. 159(4), 780–829 (2023)
Funding
Open Access funding enabled and organized by Projekt DEAL.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Köhler, K. Analytic torsion forms for fibrations by projective curves. Math. Z. 305, 47 (2023). https://doi.org/10.1007/s00209-023-03379-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00209-023-03379-3