KSemistability of cscK Manifolds with Transcendental Cohomology Class
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Abstract
We prove that constant scalar curvature Kähler (cscK) manifolds with transcendental cohomology class are Ksemistable, naturally generalising the situation for polarised manifolds. Relying on a recent result by R. Berman, T. Darvas and C. Lu regarding properness of the Kenergy, it moreover follows that cscK manifolds with discrete automorphism group are uniformly Kstable. As a main step of the proof we establish, in the general Kähler setting, a formula relating the (generalised) Donaldson–Futaki invariant to the asymptotic slope of the Kenergy along weak geodesic rays.
Keywords
Constant scalar curvature Kähler metric KStability Energy functional asymptotics YTD conjectureMathematics Subject Classification
32Q26 32Q15 14D06 53C56 14F431 Introduction
In this paper we are interested in questions of stability for constant scalar curvature Kähler (cscK) manifolds with transcendental^{1} cohomology class. To this end, let \((X,\omega )\) be a compact Kähler manifold and \(\alpha := [\omega ] \in H^{1,1}(X,\mathbb {R})\) the corresponding Kähler class. When \(\alpha \) is the first Chern class \(c_1(L)\) of some ample line bundle L over X, such questions are closely related to the Yau–Tian–Donaldson (YTD) conjecture [27, 49, 54]: A polarised algebraic manifold (X, L) is Kpolystable if and only if the polarisation class \(c_1(L)\) admits a Kähler metric of constant scalar curvature. This conjecture was recently confirmed in the Fano case, i.e. when \(L = K_X\), cf. [16, 17, 18, 52]. In this important special case, a cscK metric is nothing but a Kähler–Einstein metric. For general polarised cscK manifolds, the “if” direction of the YTD conjecture was initially proven by Mabuchi in [37], see also [5]. Prior to that, several partial results had been obtained by Donaldson [28] and Stoppa [46], both assuming that \(c_1(L)\) contains a cscK metric.
For transcendental classes very little is currently known about the validity of a correspondence between existence of cscK metrics and stability in the spirit of the YTD conjecture. Moreover, from a differential geometric point of view, there is no special reason to restrict attention to Kähler manifolds with associated integral (or rational) cohomology classes, which are then automatically of the form \(\alpha = c_1(L)\) for some ample (\(\mathbb {Q}\))line bundle L over X. In order to extend the study of stability questions to a transcendental setting, recall that there is an intersection theoretic description of the Donaldson–Futaki invariant, cf. [39, 53]. As first pointed out by Berman [4], a straightforward generalised notion of Kstability in terms of cohomology can thus be defined and a version of the YTD conjecture can be made sense of in this setting. The setup is explained in detail in Sect. 3. Our main goal is to establish the following result:
Theorem A
For precise definitions we refer to the core of the paper. As an immediate consequence of [6, Theorem 1.1] and the above Theorem A (i) we obtain the following corollary, which is a main motivation for our work (see also Remark 1.2).
Corollary 1.1
If the Kähler class \(\alpha \in H^{1,1}(X,\mathbb {R})\) admits a constant scalar curvature representative, then \((X,\alpha )\) is Ksemistable.
The corresponding statement in the case of a polarised manifold was first obtained by Donaldson in [28], as an immediate consequence of the lower bound for the Calabi functional. See also [43, 47] for related work on slope semistability. The approach taken in this paper should however be compared to, e.g. [42] and [4, 5, 12, 13], where Ksemistability is derived using so called “Kempf–Ness type” formulas. By analogy to the above papers, our proof relies on establishing such formulas valid also for transcendental classes (see Theorems B and C), in particular relating the asymptotic slope of the Kenergy along weak geodesic rays to a natural generalisation of the Donaldson–Futaki invariant. This provides a link between Ksemistability (resp. uniform Kstability) and boundedness (resp. coercivity) of the Mabuchi functional, key to establishing the stability results of Theorem A.
An underlying theme of the paper is the comparison to the extensively studied case of a polarised manifold, which becomes a “special case” in our setting. Notably, it is then known (see, e.g. [4, 5, 12, 13]) how to establish the sought Kempf–Ness type formulas using Deligne pairings; a method employed by Phong–Ross–Sturm in [42] (for further background on the Deligne pairing construction, cf. [30]). Unfortunately, such an approach breaks down in the case of a general Kähler class. In this paper, we circumvent this problem by a pluripotential approach, making use of a certain multivariate variant \(\langle \varphi _0, \dots , \varphi _n \rangle _{(\theta _0, \dots \theta _n)}\) of the Monge–Ampère energy functional, which turns out to play a role analogous to that of the Deligne pairing in arguments of the type [42]. The Deligne pairing approach should also be compared to [26, 50] using Bott–Chern forms (see, e.g. 2.4 and [44, Example 5.6]).
Remark 1.2
(Yau–Tian–Donaldson conjecture) Combining Theorem A (ii) with [22, Theorem 2.10] and a very recent result by Berman et al. [8, Theorem 1.2] we in fact further see that cscK manifolds \((X,\alpha )\) with discrete automorphism group are uniformly Kstable. The above thus confirms one direction of the Yau–Tian–Donaldson conjecture, here referring to its natural generalisation to the case of arbitrary compact Kähler manifolds with discrete automorphism group, see Sect. 5.2.
1.1 Generalised KSemistability
We briefly explain the framework we have in mind. As a starting point, there are natural generalisations of certain key concepts to the transcendental setting, a central notion being that of test configurations. First recall that a test configuration for a polarised manifold (X, L), in the sense of Donaldson, cf. [27], is given in terms of a \(\mathbb {C}^*\)equivariant degeneration \((\mathcal {X}, \mathcal {L})\) of (X, L). It can be seen as an algebrogeometric way of compactifying the product \(X \times \mathbb {C}^* \hookrightarrow \mathcal {X}\). Note that test configurations in the sense of Donaldson are now known (at least in the case of Fano manifolds, see [36]) to be equivalent to test configurations in the sense of Tian [49].
As remarked in [4], a straightforward generalisation to the transcendental setting can be given by replacing the line bundles with (1, 1)cohomology classes. In the polarised setting we would thus consider \((\mathcal {X}, c_1(\mathcal {L}))\) as a “test configuration” for \((X, c_1(L))\), by simply replacing \(\mathcal {L}\) and L with their respective first Chern classes. The details of how to formulate a good definition of such a generalised test configuration have, however, not yet been completely clarified. The definition given in this paper is motivated by a careful comparison to the usual polarised case, where we ensure that a number of basic but convenient tools still hold, cf. Sect. 3. In particular, our notion of Ksemistability coincides precisely with the usual one whenever we restrict to the case of an integral class, cf. Proposition 3.14. We will refer to such generalised test configurations as cohomological.
Definition 1.3
Remark 1.4
Note that the definition is given directly over \(\mathbb {P}^1\) so that we consider the Bott–Chern cohomology on a compact Kähler normal complex space. In the polarised case, defining a test configuration over \(\mathbb {C}\) or over \(\mathbb {P}^1\) is indeed equivalent, due to the existence of a natural \(\mathbb {C}^*\)equivariant compactification over \(\mathbb {P}^1\).
In practice, it will be enough to consider the situation when the total space \(\mathcal {X}\) is smooth and dominates \(X \times \mathbb {P}^1\), with \(\mu : \mathcal {X} \rightarrow X \times \mathbb {P}^1\) the corresponding canonical \(\mathbb {C}^*\)equivariant bimeromorphic morphism. Moreover, if \((\mathcal {X}, \mathcal {A})\) is a cohomological test configuration for \((X,\alpha )\) with \(\mathcal {X}\) as above, then \(\mathcal {A}\) is always of the form \(\mathcal {A} = \mu ^*p_1^*\alpha + [D]\), for a unique \(\mathbb {R}\)divisor D supported on the central fibre \(\mathcal {X}_0\), cf. Proposition 3.10. A cohomological test configuration can thus be characterised by an \(\mathbb {R}\)divisor, clarifying the relationship between the point of view of \(\mathbb {R}\)divisors and our cohomological approach to “transcendental Ksemistability”.
Finally, we say that \((X,\alpha )\) is Ksemistable if \(\mathrm {DF}(\mathcal {X}, \mathcal {A}) \geqslant 0\) for all cohomological test configurations \((\mathcal {X}, \mathcal {A})\) for \((X,\alpha )\) where the class \(\mathcal {A}\) is relatively Kähler, i.e. there is a Kähler form \(\beta \) on \(\mathbb {P}^1\) such that \(\mathcal {A} + \pi ^*\beta \) is Kähler on \(\mathcal {X}\). Generalised notions of (uniform) Kstability are defined analogously.
1.2 Transcendental Kempf–Ness Type Formulas
As previously stated, a central part of this paper consists in establishing a Kempf–Ness type formula connecting the Donaldson–Futaki invariant (in the sense of (1)) with the asymptotic slope of the Kenergy along certain weak geodesic rays. In fact, we first prove the following result, which is concerned with asymptotics of a certain multivariate analogue of the Monge–Ampère energy, cf. Sect. 2.2 for its definition. It turns out to be very useful for establishing a similar formula for the Kenergy (cf. Remark 1.5), but may also be of independent interest.
In what follows, we will work on the level of potentials and refer the reader to Sect. 4 for precise definitions.
Theorem B
Remark 1.5
In the setting of Hermitian line bundles, the above multivariate energy functional naturally appears as the difference (or quotient) of metrics on Deligne pairings. Moreover, note that the above theorem applies to, e.g. Aubin’s \(\mathrm {J}\)functional, the Monge–Ampère energy functional \(\mathrm {E}\) and its ‘twisted’ version \(\mathrm {E}^{\text {Ric}(\omega )}\) but not to the Kenergy \(\mathrm {M}\). Indeed, the expression for \(\mathrm {M}(\varphi _t)\) on the form \(\langle \varphi _0^t, \dots , \varphi _n^t \rangle _{(\theta _0, \dots , \theta _n)}\) involves the metric \(\log (\omega + dd^c\varphi _t)^n\) on the relative canonical bundle \(K_{\mathcal {X}/\mathbb {P}^1}\), which blows up close to \(\mathcal {X}_0\), cf. Sect. 5. As observed in [13], it is however possible to find functionals of the above form that ‘approximate’ \(\mathrm {M}\) in the sense that their asymptotic slopes coincide, up to an explicit correction term that vanishes precisely when the central fibre \(\mathcal {X}_0\) is reduced. This is a key observation.
We further remark that such a formula (2) cannot be expected to hold unless the test configurations \((\mathcal {X}_i, \mathcal {A}_i)\) and the rays \((\varphi _i^t)\) are compatible in a certain sense. This is the role of the notion of \(\mathcal {C}^{\infty }\)compatibility (as well as the \(\mathcal {C}^{1,\bar{1}}\)compatibility used in Theorem C). These notions may seem technical, but in fact mimic the case of a polarised manifold, where the situation is well understood in terms of extension of metrics on line bundles, cf. Sect. 4.1.
Theorem C
Remark 1.6
When the class \(\mathcal {A}\) on \(\mathcal {X}\) is merely relatively nef it is possible to obtain similar statements, but this necessitates much more involved arguments. Either way, the above result is more than enough for our purposes here, e.g. for proving the main result, Theorem A.
For polarised manifolds (X, L) and smooth subgeodesic rays \((\varphi _t)_{t \geqslant 0}\), this precise result was proven in [13] using Deligne pairings, as pioneered by Phong–Ross–Sturm in [42] (cf. also Paul–Tian [40, 41]). A formula in the same spirit has also been obtained for the so called Ding functional when X is a Fano variety, see [5]. However, it appears as though no version of this result was previously known in the case of nonpolarised manifolds.
1.3 Structure of the Paper
In Sect. 2 we fix our notation for energy functionals and subgeodesic rays. In particular, we introduce the multivariate energy functionals \(\langle \cdot , \dots , \cdot \rangle _{(\theta _0, \dots , \theta _n)}\), which play a central role in this paper. In Sect. 3 we introduce our generalised notion of cohomological test configurations and Ksemistability. In the case of a polarised manifold (X, L), we compare this notion to the usual algebraic one. We also discuss classes of cohomological test configurations for which it suffices to test Ksemistability, and establish a number of basic properties. In Sect. 4 we discuss transcendental Kempf–Nesstype formulas and prove Theorem B. This involves introducing natural compatibility conditions between a ray \((\varphi _t)\) and a cohomological test configuration \((\mathcal {X}, \mathcal {A})\) for \((X,\alpha )\). As a useful special case, we discuss the weak geodesic ray associated with \((\mathcal {X}, \mathcal {A})\). In Sect. 5 we finally apply Theorem B to yield a weak version of Theorem C, from which we in turn deduce our main result, Theorem A. By an immediate adaptation of techniques from [13] we then compute the precise asymptotic slope of the Mabuchi functional, thus establishing the full Theorem C.
2 Preliminaries
2.1 Notation and Basic Definitions
Let X be a compact complex manifold of \(\mathrm {dim}_{\mathbb {C}}X = n\) equipped with a given Kähler form \(\omega \), i.e. a smooth real closed positive (1, 1)form on X. Denote the Kähler class \([\omega ] \in H^{1,1}(X,\mathbb {R})\) by \(\alpha \).
2.2 Energy Functionals and a Deligne Functional Formalism
 symmetric, i.e. for any permutation \(\sigma \) of the set \(\{0,1, \dots ,n\}\), we have$$\begin{aligned} \langle \varphi _{\sigma (0)}, \ldots , \varphi _{\sigma (n)} \rangle _{(\theta _{\sigma (0)}, \ldots , \theta _{\sigma (n)})} = \langle \varphi _0, \ldots , \varphi _n \rangle _{(\theta _0,\dots ,\theta _n)}. \end{aligned}$$
 if \(\varphi _0'\) is another \(\theta _i\)psh function in \(\mathrm {PSH}(X,\theta ) \cap L^{\infty }_{\mathrm {loc}}\), then we have a ‘change of function’ property$$\begin{aligned}&\langle \varphi _0', \varphi _1 \dots , \varphi _n \rangle  \langle \varphi _0, \varphi _1 \dots , \varphi _n \rangle \\&\quad =\int _X (\varphi _0'  \varphi _0) \; (\omega _1 + dd^c\varphi _1) \wedge \dots \wedge (\omega _n + dd^c\varphi _n). \end{aligned}$$
Definition 2.1
Remark 2.2
The multivariate energy functional \(\langle \cdot , \dots , \cdot \rangle _{(\theta _0,\dots ,\theta _n)}\) can also be defined on \(\mathcal {C}^{\infty }(X) \times \dots \times \mathcal {C}^{\infty }(X)\) by the same formula. In Sects. 4 and 5 it will be interesting to consider both the smooth case and the case of locally bounded \(\theta _i\)psh functions.
Using integration by parts one can check that this functional is indeed symmetric.
Proposition 2.3
The functional \(\langle \cdot , \dots , \cdot \rangle _{(\theta _0,\dots ,\theta _n)}\) is symmetric.
Proof
Example 2.4
2.3 Subgeodesic Rays
We will use the following standard terminology, motivated by the extensive study of (weak) geodesics in the space \(\mathcal {H}\), see, e.g. [9, 15, 20, 27, 45].
Definition 2.5
Definition 2.6
Viewing the family \((\varphi _t)_{t \geqslant 0}\) as a map \((0,+\infty ) \rightarrow \mathrm {PSH}(X,\omega )\), we say that \((\varphi ^t)_{t \geqslant 0}\) is continuous (resp. locally bounded, smooth) if the corresponding \(S^1\)invariant function \(\Phi \) is continuous (resp. locally bounded, smooth).
The existence of geodesics with bounded Laplacian was proven by Chen [15] with complements by Blocki [9], see also, e.g. [20, 21]. We will refer to such a geodesic as being \(\mathcal {C}^{1,\bar{1}}\)regular, cf. Lemma 4.6.
Definition 2.7
We say that a function \(\varphi \) is \(\mathcal {C}^{1,\bar{1}}\)regular if \(dd^c\varphi \in L^{\infty }_{\mathrm {loc}}\), and we set \(\mathcal {H}^{1,\bar{1}}:= \mathrm {PSH}(X,\omega ) \cap \mathcal {C}^{1,\bar{1}}\).
Recall that a \(\mathcal {C}^{1,\bar{1}}\)regular function is automatically \(\mathcal {C}^{1,a}\)regular for all \(0< a < 1\). On the other hand, this condition is weaker than \(\mathcal {C}^{1,1}\)regularity (i.e. bounded real Hessian).
2.4 SecondOrder Variation of Deligne Functionals
We have the following identity for the secondorder variations of the multivariate energy functional \(\langle \cdot , \dots , \cdot \rangle _{(\theta _0, \dots , \theta _n)}\).
Proposition 2.8
Proof
The result follows from a computation relying on integration by parts and is an immediate adaptation of, for instance, [7, Proposition 6.2]. \(\square \)
2.5 The KEnergy and the Chen–Tian Formula
Following Chen [14] (using the formula (5)) we will often work with the extension \(\mathrm {M}:\mathcal {H}^{1,\bar{1}} \rightarrow \mathbb {R}\) of the Mabuchi functional to the space of \(\omega \)psh functions with bounded Laplacian. This is a natural setting to consider, since weak geodesic rays with bounded Laplacian are known to always exist, cf. [9, 15, 20, 21] as well as Lemma 4.6.
For later use, we also state the following definition.
Definition 2.9
We further recall that the Mabuchi functional is convex along weak geodesic rays, as was recently established by [6], see also [19]. As a consequence of this convexity, the Mabuchi functional is bounded from below (in the given Kähler class) whenever \(\alpha \) contains a cscK metric, see [29, 35] for a proof in the polarised case and [6] for the general Kähler setting.
3 Cohomological Test Configurations and KSemistability
In this section we introduce a natural generalised notion of test configurations and Ksemistability of \((X,\alpha )\) that has meaning even when the class \(\alpha \in H^{1,1}(X,\mathbb {R})\) is nonintegral (or nonrational), i.e. when \(\alpha \) is not necessarily of the form \(c_1(L)\) for some ample (\(\mathbb {Q}\))line bundle L on X. As remarked by Berman in [4], it is natural to generalise the notion of test configuration in terms of cohomology classes. In the polarised setting, the idea is to consider \((\mathcal {X}, c_1(\mathcal {L}))\) as a “test configuration” for \((X, c_1(L))\), by simply replacing \(\mathcal {L}\) and L with their respective first Chern classes. This approach is motivated in detail below. Moreover, a number of basic and useful properties will be established, and throughout, this generalisation will systematically be compared to the original notion of algebraic test configuration \((\mathcal {X}, \mathcal {L})\) for (X, L), introduced by Donaldson in [27].
Remark 3.1
Much of the following exposition goes through even when the cohomology class \(\alpha \) is not Kähler. Unless explicitly stated otherwise, we thus assume that \(\alpha = [\theta ]\) for some closed (1, 1)form \(\theta \) on X.
3.1 Test Configurations for X
We first introduce the notion of test configuration \(\mathcal {X}\) for X, working directly over \(\mathbb {P}^1\). For the sake of comparison, recall the usual concept of test configuration for polarised manifolds, see, e.g. [12, 48]. In what follows, we refer to [31] for background on normal complex spaces.
Definition 3.2

a normal compact Kähler complex space \(\mathcal {X}\) with a flat morphism \(\pi : \mathcal {X} \rightarrow \mathbb {P}^1\)

a \(\mathbb {C}^*\)action \(\lambda \) on \(\mathcal {X}\) lifting the canonical action on \(\mathbb {P}^1\)
 a \(\mathbb {C}^*\)equivariant isomorphism$$\begin{aligned} \mathcal {X} {\setminus } \mathcal {X}_0 \simeq X \times (\mathbb {P}^1 {\setminus } \{0\}). \end{aligned}$$(6)
The isomorphism 6 gives an open embedding of \(X \times (\mathbb {P}^1 {\setminus } \{0\})\) into \(\mathcal {X}\), hence induces a canonical \(\mathbb {C}^*\)equivariant bimeromorphic map \(\mu : \mathcal {X} \dashrightarrow X \times \mathbb {P}^1\). We say that \(\mathcal {X}\) dominates \(X \times \mathbb {P}^1\) if the above bimeromorphic map \(\mu \) is a morphism. Taking \(\mathcal {X}'\) to be the normalisation of the graph of \(\mathcal {X} \dashrightarrow X \times \mathbb {P}^1\) we obtain a \(\mathbb {C}^*\)equivariant bimeromorphic morphism \(\rho : \mathcal {X}' \rightarrow \mathcal {X}\) with \(\mathcal {X}'\) normal and dominating \(X \times \mathbb {P}^1\). In the terminology of [12] such a morphism \(\rho \) is called a determination of \(\mathcal {X}\). In particular, a determination of \(\mathcal {X}\) always exists. By the above considerations we will often, up to replacing \(\mathcal {X}\) by \(\mathcal {X}'\), be able to assume that the given test configuration for X dominates \(X \times \mathbb {P}^1\).
Moreover, any test configuration \(\mathcal {X}\) for X can be dominated by a smooth test configuration \(\mathcal {X}'\) for X (where we may even assume that \(\mathcal {X}'_0\) is a divisor of simple normal crossings). Indeed, by Hironaka (see [33, Theorem 45] for the precise statement concerning normal complex spaces) there is a \(\mathbb {C}^*\)equivariant proper bimeromorphic map \(\mu : \mathcal {X}' \rightarrow \mathcal {X}\), with \(\mathcal {X}'\) smooth, such that \(\mathcal {X}_0'\) has simple normal crossings and \(\mu \) is an isomorphism outside of the central fibre \(\mathcal {X}_0\).
As a further consequence of the isomorphism (6), note that if \(\Phi \) is a function on \(\mathcal {X}\), then its restriction to each fibre \(\mathcal {X}_{\tau } \simeq X\), \(\tau \in \mathbb {P}^1 {\setminus } \{0\}\) identifies with a function on X. The function \(\Phi \) thus gives rise to a family of functions \((\varphi _t)_{t \geqslant 0}\) on X, recalling our convention of reparametrising so that \(t :=  \log \tau \).
Remark 3.3
When X is projective (hence algebraic), the GAGA principle shows that the usual (i.e. algebraic, and normal) test configurations of X correspond precisely to the test configurations (in our sense of Definition 3.2) with \(\mathcal {X}\) projective.
3.2 Cohomological Test Configurations for \((X,\alpha )\)
We now introduce a natural generalisation of the usual notion of algebraic test configuration\((\mathcal {X}, \mathcal {L})\) for a polarised manifold (X, L). This following definition involves the Bott–Chern cohomology on normal complex spaces, i.e. the space of locally \(dd^c\)exact (1, 1)forms (or currents) modulo globally \(dd^c\)exact (1, 1)forms (or currents). The Bott–Chern cohomology is finite dimensional and the cohomology classes can be pulled back. Moreover, \(H^{1,1}_{\mathrm {BC}}(\mathcal {X}, \mathbb {R})\) coincides with the usual Dolbeault cohomology \(H^{1,1}(\mathcal {X},\mathbb {R})\) whenever \(\mathcal {X}\) is smooth. See, e.g. [11] for background.
Definition 3.4
Definition 3.5
We say that a test configuration \((\mathcal {X}, \mathcal {A})\) for \((X,\alpha )\) is smooth if the total space \(\mathcal {X}\) is smooth. In case \(\alpha \in H^{1,1}(X,\mathbb {R})\) is Kähler, we say that \((\mathcal {X}, \mathcal {A})\) is relatively Kähler if the cohomology class \(\mathcal {A}\) is relatively Kähler, i.e. there is a Kähler form \(\beta \) on \(\mathbb {P}^1\) such that \(\mathcal {A} + \pi ^*\beta \) is Kähler on \(\mathcal {X}\).
Exploiting the discussion following Definition 3.2 we in practice restrict attention to the situation when \((\mathcal {X}, \mathcal {A})\) is a smooth (cohomological) test configuration for \((X,\alpha )\) dominating \(X \times \mathbb {P}^1\), with \(\mu : \mathcal {X} \rightarrow X \times \mathbb {P}^1\) the corresponding \(\mathbb {C}^*\)equivariant bimeromorphic morphism. This situation is studied in detail in Sect. 3.4, where we in particular show that the class \(\mathcal {A} \in H^{1,1}(\mathcal {X}, \mathbb {R})\) is always of the form \(\mathcal {A} = \mu ^*p_1^*\alpha + [D]\) for a unique \(\mathbb {R}\)divisor D supported on the central fibre, cf. Proposition 3.10.
It is further natural to ask how the above notion of cohomological test configurations compares to the algebraic test configurations introduced by Donaldson in [27]. On the one hand, we have the following example:
Example 3.6
If \((\mathcal {Y}, \mathcal {L})\) is an algebraic test configuration for (X, L) and we let \(\bar{\mathcal {Y}}\), \(\bar{\mathcal {L}}\) and \(\bar{L}\), respectively, denote the \(\mathbb {C}^*\)equivariant compactifications over \(\mathbb {P}^1\), then \((\bar{\mathcal {Y}}, c_1(\mathcal {L}))\) is a cohomological test configuration for \((X, c_1(L))\), canonically induced by \((\mathcal {Y}, \mathcal {L})\).
On the other hand, there is no converse such correspondence. For instance, even if (X, L) is a polarised manifold there are more cohomological test configurations \((\mathcal {X},\mathcal {A})\) for \((X,c_1(L))\) than algebraic test configurations \((\mathcal {Y}, \mathcal {L})\) for (X, L). However, we show in Proposition 3.14 that such considerations are not an issue in the study of Ksemistability of \((X,\alpha )\).
3.3 The Donaldson–Futaki Invariant and KSemistability
The following generalisation of the Donaldson–Futaki invariant is straightforward, at least when the test configuration is smooth (in general one can use resolution of singularities to make sense of the intersection number below).
Definition 3.7
We recall that \(\mathcal {X}\) is assumed to be compact, cf. Definition 3.2, and that \(K_{\mathcal {X}/\mathbb {P}^1} := K_{\mathcal {X}}  \pi ^*K_{\mathbb {P}^1}\) denotes the relative canonical divisor. The point is that by results of Wang [53] and Odaka [39] \(\mathrm {DF}(\bar{\mathcal {Y}},c_1(\mathcal {L}))\) coincides with \(\mathrm {DF}(\mathcal {Y},\mathcal {L})\) whenever \((\mathcal {Y},\mathcal {L})\) is an algebraic test configuration for a polarised manifold (X, L), with \(\mathcal {Y}\) normal (see the proof of Proposition 3.14). Hence the above quantity is a generalisation of the classical Donaldson–Futaki invariant.
The analogue of Ksemistability in the context of cohomological test configurations is defined as follows.
Definition 3.8
We say that \((X,\alpha )\) is Ksemistable if \(\mathrm {DF}(\mathcal {X},\mathcal {A}) \geqslant 0\) for all relatively Kähler test configurations \((\mathcal {X},\mathcal {A})\) for \((X,\alpha )\).
Remark 3.9
With the study of Ksemistability in mind, we emphasise that the Donaldson–Futaki invariant \(\mathrm {DF}(\mathcal {Y},\mathcal {L})\) (cf. [39, 53]) depends only on \(\mathcal {Y}\) and \(c_1(\mathcal {L})\). The notion of cohomological test configuration emphasises this fact.
In order to further motivate the above definitions, we now introduce a number of related concepts and basic properties that will be useful in the sequel.
3.4 Test Configurations Characterised by \(\mathbb {R}\)Divisors
Recall that if \((\mathcal {X},\mathcal {L})\) is an algebraic test configuration for a polarised manifold (X, L) that dominates \((X,L) \times \mathbb {C}\), then \(\mathcal {L} = \mu ^* p_1^*L + D\) for a unique \(\mathbb {Q}\)Cartier divisor D supported on \(\mathcal {X}_0\), see [12]. Similarly, the following result characterises the classes \(\mathcal {A}\) associated with smooth and dominating cohomological test configurations, in terms of \(\mathbb {R}\)divisors D supported on the central fibre \(\mathcal {X}_0\).
Proposition 3.10
Proof
Let \(\alpha := [\omega ] \in H^{1,1}(X,\mathbb {R})\). We begin by proving existence: By hypothesis \(\mathcal {X}\) dominates \(X \times \mathbb {P}^1\) via the morphism \(\mu \), such that the central fibre decomposes into the strict transform of \(X \times \{0\}\) and the \(\mu \)exceptional divisor. We write \(\mathcal {X}_0 = \sum _i b_i E_i\), with \(E_i\) irreducible. Denoting by [E] the cohomology class of E and by \(p_1: X \times \mathbb {P}^1 \rightarrow X\) the projection on the first factor, we then have the following formula: \(\square \)
Lemma 3.11
\(H^{1,1}(\mathcal {X}) = \mu ^*p_1^*H^{1,1}(X) \; \oplus \; \bigoplus _i \mathbb {R} [E_i]\).
Proof of Lemma 3.11
Let \(\Theta \) be a closed (1, 1)form on \(\mathcal {X}\). Then \(T := \Theta  \mu ^*(\mu _*\Theta )\) is a closed (1, 1)current of order 0 supported on \(\cup _i E_i = \mathrm {Exc}(\mu )\). By Demailly’s second theorem of support (see [23]) it follows that \(T = \sum _i \lambda _i \delta _{E_j}\) and hence \([\Theta ] = \mu ^*(\mu _*[\Theta ]) + \sum _i \lambda _i [E_i]\) in \(H^{1,1}(\mathcal {X})\).
Since \(H^{1,1}(\mathbb {P}^1)\) is generated by [0], we have \(p_2^*H^{1,1}(\mathbb {P}^1) = \mathbb {R}[X \times \{0\}]\). By the Künneth formula, it thus follows that \(H^{1,1}(\mathcal {X}) = \mu ^*p_1^*H^{1,1}(X) \; \oplus \; \mu ^*(\mathbb {R}[X \times \{0\}])\; \oplus \; \bigoplus _i \mathbb {R} [E_i]\). \(\square \)
If we decompose \(\mathcal {A}\) accordingly we obtain \(\mathcal {A} = \mu ^*p_1^*\eta + [D]\) with \(D := \mu ^*(c[X \times \{0\}]) + \bigoplus _i b_i [E_i]\) and \(\eta \) a class in \(H^{1,1}(X)\). The restrictions of \(\mathcal {A}\) and \(\mu ^*p_1^*\alpha \) to \(\pi ^{1}(1) \simeq X \times \{1\} \simeq X\) are identified with \(\alpha \) and \(\eta \), respectively. Since D is supported on \(\mathcal {X}_0\) it follows that \(\eta = \alpha \). We thus have the sought decomposition, proving existence.
This gives a very convenient characterisation of smooth cohomological test configurations for \((X,\alpha )\) that dominate \(X \times \mathbb {P}^1\).
Proposition 3.12
Let \(\alpha \in H^{1,1}(X,\mathbb {R})\) be Kähler. Then \((X,\alpha )\) is Ksemistable if and only if \(\mathrm {DF}(\mathcal {X},\mathcal {A}) \geqslant 0\) for all smooth, relatively Kähler cohomological test configurations \((\mathcal {X},\mathcal {A})\) for \((X, \alpha )\) dominating \(X \times \mathbb {P}^1\).
Proof
Remark 3.13
With respect to testing Ksemistability one can in fact restrict the class of test configurations that need to be considered even further, as explained in Sect. 3.6.
3.5 Cohomological KSemistability for Polarised Manifolds
It is useful to compare cohomological and algebraic Ksemistability in the special case of a polarised manifold (X, L).
Proposition 3.14
Let (X, L) be a polarised manifold and let \(\alpha := c_1(L)\). Then \((X,c_1(L))\) is (cohomologically) Ksemistable if and only if (X, L) is (algebraically) Ksemistable.
Proof
Suppose that \((X, c_1(L))\) is cohomologically Ksemistable. If \((\mathcal {X}, \mathcal {L})\) is an ample test configuration for (X, L), let \(\mathcal {A} := c_1(\bar{\mathcal {L}})\). By the intersection theoretic characterisation of the Donaldson–Futaki invariant (Definition 3.7) we then have \(\mathrm {DF}(\mathcal {X},\mathcal {A}) = \mathrm {DF}(\mathcal {X},\mathcal {L}) \geqslant 0\). Hence (X, L) is algebraically Ksemistable.
3.6 The NonArchimedean Mabuchi Functional and Base Change
Let \((\mathcal {X},\mathcal {A})\) be a cohomological test configuration for \((X,\alpha )\). A natural operation on \((\mathcal {X}, \mathcal {A})\) is that of base change (on \(\mathcal {X}\) and we pull back \(\mathcal {A}\)). Unlike resolution of singularities, however, the \(\mathrm {DF}\)invariant does not behave well under base change. In this context, a more natural object of study is instead the nonArchimedean Mabuchi functional\(\mathrm {M}^{\mathrm {NA}}\) (first introduced in [12, 13], where also an explanation of the terminology is given).
Definition 3.15
Note that the ‘correction term’ \(V^{1}((\mathcal {X}_{0,\mathrm {red}}  \mathcal {X}_0) \cdot \mathcal {A}^n)_{\mathcal {X}}\) is nonpositive and vanishes precisely when the central fibre \(\mathcal {X}_0\) is reduced. The point of adding to \(\mathrm {DF}\) this additional term is that the resulting quantity \(\mathrm {M}^{\mathrm {NA}}(\mathcal {X}, \mathcal {A})\) becomes homogeneous under base change, i.e. we have the following lemma.
Lemma 3.16
Proof
We refer the reader to [12, Proposition 7.13], whose proof goes through in the analytic case as well. \(\square \)
Proposition 3.17
Let \(\alpha \in H^{1,1}(X,\mathbb {R})\) be Kähler. Then \((X,\alpha )\) is Ksemistable (Definition 3.8) if and only if \(\mathrm {DF}(\mathcal {X}, \mathcal {A}) \geqslant 0\) for all semistable, relatively Kähler cohomological test configurations \((\mathcal {X}, \mathcal {A})\) for \((X, \alpha )\) dominating \(X \times \mathbb {P}^1\).
4 Transcendental Kempf–Ness Type Formulas
Let X be a compact Kähler manifold of dimension n and let \(\theta _i\), \(0 \leqslant i \leqslant n\), be closed (1, 1)forms on X. Let \(\alpha _i := [\theta _i] \in H^{1,1}(X,\mathbb {R})\) be the corresponding cohomology classes. In this section we aim to prove Theorem B. In other words, we establish a Kempf–Ness type formula (for cohomological test configurations), which connects the asymptotic slope of the multivariate energy functional \(\langle \varphi _0^t,\dots ,\varphi _n^t \rangle _{(\theta _0, \dots , \theta _n)}\) (see Definition 2.2) with a certain intersection number. In order for such a result to hold, we need to ask that the rays \((\varphi _i^t)_{t \geqslant 0}\) are compatible with \((\mathcal {X}_i, \mathcal {A}_i)\) in a sense that has to do with extension across the central fibre, see Sect. 4.1.
For what follows, note that, by equivariant resolution of singularities, there is a test configuration \(\mathcal {X}\) for X which is smooth and dominates \(X \times \mathbb {P}^1\). This setup comes with canonical\(\mathbb {C}^*\)equivariant bimeromorphic maps \(\rho _i: \mathcal {X} \rightarrow \mathcal {X}_i\), respectively. In particular:
Definition 4.1
Remark 4.2
Up to desingularising we can and we will in this section consider only smooth cohomological test configurations \((\mathcal {X}_i,\mathcal {A}_i)\) for \((X, \alpha _i)\) dominating \(X \times \mathbb {P}^1\), with \(\mu _i: \mathcal {X}_i \rightarrow X \times \mathbb {P}^1\) the corresponding \(\mathbb {C}^*\)equivariant bimeromorphic morphisms, respectively. We content ourselves by noting that the following \(\mathcal {C}^{\infty }\)compatibility condition can be defined (much in the same way, using a desingularisation) in the singular case as well.
4.1 Compatibility of Rays and Test Configurations
The main purpose of this section is to establish Theorem B, which is a formula relating algebraic (intersection theoretic) quantities to asymptotic slopes of Deligne functionals (e.g. \(\mathrm {E}\) or \(\mathrm {J}\)) along certain rays. However, such a formula cannot hold for any such ray. The point of the following compatibility conditions is to establish some natural situations in which this formula holds. Technically, recall that a ray \((\varphi _t)_{t \geqslant 0}\) on X is in correspondence with an \(S^1\)invariant functions \(\Phi \) on \(X \times \bar{\Delta }^*\). The proof of Theorem B, will show that it is important to extend the function \(\Phi \circ \mu \) on \(\mathcal {X} {\setminus } \mathcal {X}_0\) also across the central fibre \(\mathcal {X}_0\).
 (i)
smooth but not necessarily subgeodesic rays \((\varphi _t)\) that are \(\mathcal {C}^{\infty }\)compatible with the smooth test configuration \((\mathcal {X}, \mathcal {A})\) for \((X,\alpha )\), dominating \(X \times \mathbb {P}^1\). Here we can consider \(\alpha = [\theta ] \in H^{1,1}(X,\mathbb {R})\) for any closed (1, 1)form \(\theta \) on X.
 (ii)
locally bounded subgeodesic rays \((\varphi _t)\) that are \(L^{\infty }\)compatible or (more restrictively) \(\mathcal {C}^{1,\bar{1}}\)compatible with the given smooth and relatively Kähler test configuration \((\mathcal {X}, \mathcal {A})\) for \((X,\alpha )\), dominating \(X \times \mathbb {P}^1\). Here we thus suppose that \(\alpha \) is a Kähler class.
4.2 \(\mathcal {C}^{\infty }\)Compatible Rays
We first introduce the notion of smooth (not necessarily subgeodesic) rays that are \(\mathcal {C}^{\infty }\)compatible with the given test configuration \((\mathcal {X}, \mathcal {A})\) for \((X,\alpha )\).
Definition 4.3
Let \((\varphi _t)_{t \geqslant 0}\) be a smooth ray in \(\mathcal {C}^{\infty }(X)\) and denote by \(\Phi \) the corresponding smooth \(S^1\)invariant function on \(X \times \bar{\Delta }^*\). We say that \((\varphi _t)\) and \((\mathcal {X},\mathcal {A})\) are \(\mathcal {C}^{\infty }\)compatible if \(\Phi \circ \mu +\psi _D\) extends smoothly across \(\mathcal {X}_0\).
The condition is indeed independent of the choice of \(\psi _D\), as the latter is a welldefined modulo smooth function. In the case of a polarised manifold (X, L) with an (algebraic) test configuration \((\mathcal {X}, \mathcal {L})\) this condition amounts to demanding that the metric on \(\mathcal {L}\) associated with the ray \((\varphi _t)_{t \geqslant 0}\) extends smoothly across the central fibre.
As a useful ‘model example’ to keep in mind, let \(\Omega \) be a smooth \(S^1\)invariant representative of \(\mathcal {A}\) and denote the restrictions \(\Omega _{\vert \mathcal {X}_{\tau }} =: \Omega _{\tau }\). Note that \(\Omega _{\tau }\) and \(\Omega _1\) are cohomologous for each \(\tau \in \mathbb {P}^1 {\setminus } \{0\}\), and hence we may define a ray \((\varphi _t)_{t \geqslant 0}\) on X, \(\mathcal {C}^{\infty }\)compatible with \((\mathcal {X}, \mathcal {A})\), by the following relation \(\lambda (\tau )^*\Omega _{\tau }  \Omega _1 = dd^c\varphi _{\tau }\), where \(t = \log \tau \) and \(\lambda (\tau ): \mathcal {X}_{\tau } \rightarrow \mathcal {X}_1 \simeq X\) is the isomorphism induced by the \(\mathbb {C}^*\)action \(\lambda \) on \(\mathcal {X}\).
We further establish existence of a smooth \(\mathcal {C}^{\infty }\)compatible subgeodesic ray associated to a given relatively Kähler test configuration \((\mathcal {X}, \mathcal {A})\) for \((X,\alpha )\).
Lemma 4.4
If \(\mathcal {A}\) is relatively Kähler, then \((\mathcal {X},\mathcal {A})\) is \(\mathcal {C}^{\infty }\)compatible with some smooth subgeodesic ray \((\varphi _t)\).
Proof
4.3 \(\mathcal {C}^{1,\bar{1}}\)Compatible Rays and the Weak Geodesic Ray Associated with \((\mathcal {X}, \mathcal {A})\)
Let \((\mathcal {X}, \mathcal {A})\) be a smooth, relatively Kähler cohomological test configuration for \((X,\alpha )\) (with \(\alpha \) Kähler). With this setup, it is also interesting to consider the following weaker compatibility conditions, referred to as \(L^{\infty }\)compatibility and \(\mathcal {C}^{1,\bar{1}}\)compatibility, respectively.
Definition 4.5
Let \((\varphi _t)_{t \geqslant 0}\) be a locally bounded subgeodesic ray, and denote by \(\Phi \) the corresponding \(S^1\)invariant locally bounded \(p_1^*\omega \)psh function on \(X\times \bar{\Delta }^*\). We say that \((\varphi _t)\) and \((\mathcal {X},\mathcal {A})\) are \(L^{\infty }\)compatible if \(\Phi \circ \mu +\psi _D\) is locally bounded near \(\mathcal {X}_0\), resp. \(\mathcal {C}^{1,\bar{1}}\)compatible if \(\Phi \circ \mu +\psi _D\) is of class \(\mathcal {C}^{1,\bar{1}}\) on \(\pi ^{1}(\Delta )\).
Indeed, we will see that \(\mathcal {C}^{1,\bar{1}}\)compatibility is always satisfied for weak geodesic rays associated with \((\mathcal {X}, \mathcal {A})\). In particular, for any given test configuration, \(\mathcal {C}^{1,\bar{1}}\)compatible subgeodesics always exist. This is the content of the following result, which is a consequence of the theory for degenerate Monge–Ampère equations on manifolds with boundary. We refer the reader to [10] for the relevant background.
Lemma 4.6
With the situation (2) in mind, let \((\mathcal {X},\mathcal {A})\) be a smooth, relatively Kähler cohomological test configuration of \((X,\alpha )\) dominating \(X \times \mathbb {P}^1\). Then \((\mathcal {X}, \mathcal {A})\) is \(\mathcal {C}^{1,\bar{1}}\)compatible with some weak geodesic ray \((\varphi _t)_{t \geqslant 0}\).
Remark 4.7
The proof will show that the constructed ray is actually unique, once a \(\varphi _0 \in \mathcal {H}\) is fixed.
Proof of Lemma 4.6
Let \(M := \pi ^{1}(\bar{\Delta }) \subset \mathcal {X} \). It is a smooth complex manifold with boundary \(\partial M = \pi ^{1}(S^1)\).
4.4 A Useful Lemma
We now note that in order to compute the asymptotic slope of the Monge–Ampère energy functional \(\mathrm {E}\) or its multivariate analogue \(\mathrm {E}_{(\omega _0, \dots , \omega _n)}\) we may in fact replace \(L^{\infty }\)compatible rays \((\varphi ^t)\) with \((\mathcal {X}, \mathcal {A})\) by \(\mathcal {C}^{\infty }\)compatible ones. Indeed, note that any two locally bounded subgeodesic rays \((\varphi _t)\) and \((\varphi '_t)\)\(L^{\infty }\)compatible with \((\mathcal {X},\mathcal {A})\) satisfy \(\Phi \circ \mu =\Phi '\circ \mu +O(1)\) near \(\mathcal {X}_0\), and hence \(\varphi _t=\varphi '_t+O(1)\) as \(t\rightarrow +\infty \). This leads to the following observation, which will be useful in the view of proving Theorems B and C.
Lemma 4.8
Proof
4.5 Asymptotic Slope of Deligne Functionals: Proof of Theorem B
With the above formalism in place, we are ready to formulate the main result of this section (Theorem B of the introduction). It constitutes the main contribution towards establishing Theorem A, and may be viewed as a transcendental analogue of Lemma 4.3 in [13]. We here formulate and prove the theorem in the ‘smooth but not necessarily Kähler’ setting (see Sect. 4.1, situation (1)). However, one should note that there is also a valid formulation for \(L^{\infty }\)compatible subgeodesics, as pointed out in Remark 4.11.
Theorem 4.9
Proof
Fix any smooth \(S^1\)invariant (1, 1)forms \(\Omega _i\) on \(\mathcal {X}_i\) such that \([\Omega _i] =\mathcal {A}_i\) in \(H^{1,1}(\mathcal {X}_i, \mathbb {R})\). Let \((\varphi _i^t)_{t\geqslant 0}\) be smooth and \(\mathcal {C}^{\infty }\)compatible with \((\mathcal {X}_i,\mathcal {A}_i)\), respectively. Let \(\mathcal {X}\) be a smooth test configuration that simultaneously dominates the \(\mathcal {X}_i\). By pulling back to \(\mathcal {X}\) we can assume that the \(\mathcal {X}_i\) are all equal (note that the notion of being \(\mathcal {C}^{\infty }\)compatible is preserved under this pullback).
In the notation of Sect. 4.1, the functions \(\Phi _i\circ \mu +\psi _D\) are then smooth on the manifold with boundary \(M:=\pi ^{1}(\bar{\Delta })\), and may thus be written as the restriction of smooth \(S^1\)invariant functions \(\Psi _i\) on \(\mathcal {X}\), respectively.
Using the \(\mathbb {C}^*\)equivariant isomorphism \(\mathcal {X}{\setminus }\mathcal {X}_0\simeq X\times (\mathbb {P}^1{\setminus }\{0\})\) we view \((\Psi _i\psi _D)_{ \mathcal {X}_{\tau }}\) as a function \(\varphi _i^{\tau }\in \mathcal {C}^{\infty }(X)\). By Proposition 2.8 we then have \(\square \)
Lemma 4.10
Proof
The result follows from Proposition 2.8 and the fact that \(\mu \) is a biholomorphism away from \(\tau = 0\), where also \(\delta _D = 0\) (recalling that the \(\mathbb {R}\)divisor D is supported on \(\mathcal {X}_0\)). \(\square \)
Remark 4.11
The above proof in fact also yields a version of Theorem 4.9 for subgeodesics\((\varphi _i^t)_{t \geqslant 0}\) that are \(L^{\infty }\)compatible with smooth test configurations \((\mathcal {X}_i, \mathcal {A}_i)\) for \((X,\alpha _i)\) dominating \(X \times \mathbb {P}^1\). This follows from the observation that one may replace \(L^{\infty }\)compatible subgeodesic rays with smooth \(\mathcal {C}^{\infty }\)compatible ones, using Lemmas 4.4 and 4.8.
Corollary 4.12
Remark 4.13
Here \(\mathrm {E}^{\mathrm {NA}}\) makes reference to the nonArchimedean Monge–Ampère energy functional, see [12] for an explanation of the terminology.
To give a second example of an immediate corollary, interesting in its own right, we state the following (compare [25]):
Corollary 4.14
Proof
Note that we may write \(\mathrm {J}(\varphi _t) = V^{1}\langle \varphi _t, 0,\dots ,0 \rangle _{(\omega , \dots , \omega )}  \mathrm {E}(\varphi _t)\) and apply Theorem 4.9. \(\square \)
5 Asymptotics for the KEnergy
Let \((X,\omega )\) be a compact Kähler manifold and \(\alpha := [\omega ] \in H^{1,1}(X,\mathbb {R})\) a Kähler class on X. As before, let \((\mathcal {X}, \mathcal {A})\) be a smooth, relatively Kähler cohomological test configuration for \((X, \alpha )\) dominating \(X \times \mathbb {P}^1\). In this section we explain how the above Theorem 4.9 can be used to compute the asymptotic slope of the Mabuchi (Kenergy) functional along rays \((\varphi ^t)\), \(\mathcal {C}^{1,\bar{1}}\)compatible with \((\mathcal {X},\mathcal {A})\). It is useful to keep the case of weak geodesic rays (as constructed in Lemma 4.6) in mind, which in turn implies Ksemistability of \((X,\alpha )\) (Theorem A).
5.1 A Weak Version of Theorem C
We first explain how to obtain a weak version of Theorem C, as a direct consequence of Theorem 4.9. This version is more direct to establish than the full Theorem C, and will in fact be sufficient in order to prove Ksemistability of \((X,\alpha )\), as explained in Sect. 5.2.
Theorem 5.1
Remark 5.2
In view of the strong version (see Theorem 5.6) we actually know that the limit is well defined and, moreover, we obtain this way the precise asymptotic slope of the Mabuchi functional, see Sect. 5.3.
Proof of Theorem 5.1
Lemma 5.3
Proof
This result is an immediate consequence of (8), the Chen–Tian formula (5) and Corollary 4.12. \(\square \)
In view of Proposition 3.10, we as usual let D denote the unique \(\mathbb {R}\)divisor supported on \(\mathcal {X}_0\) such that \( \mathcal {A}=\mu ^*p_1^*\alpha +[D], \) with \(p_1:X \times \mathbb {P}^1 \rightarrow X\) the first projection. Fix a choice of an \(S^1\)invariant function ‘Green function’ \(\psi _D\) for D, so that \(\delta _D=\theta _D+dd^c\psi _D\) with \(\theta _D\) a smooth \(S^1\)invariant closed (1, 1)form on \(\mathcal {X}\). Moreover, set \(\Omega := \mu ^*p_1^*\alpha + \theta _D\) (for which \([\Omega ] = \mathcal {A}\) then holds) and let \(\Phi \) denote the \(S^1\)invariant function on \(X \times \mathbb {P}^1\) corresponding to the ray \((\varphi _t)\). In particular, the function \(\Phi \circ \mu + \psi _D\) extends to a smooth \(\Omega \)psh function \(\Psi \) on \(\mathcal {X}\), by \(\mathcal {C}^{\infty }\)compatibility.
As explained below, the above ‘weak Theorem C’ actually suffices to yield our main result.
5.2 Proof of Theorem A
We now explain how the above considerations apply to give a proof of Theorem A and point out some immediate and important consequences regarding the YTD conjecture. First recall the following definition (see, e.g. [51, Sect. 7.2]):
Definition 5.4
We are now ready to prove Theorem A.
Proof of Theorem A
As remarked in the introduction it follows from convexity of the Mabuchi functional along weak geodesic rays, cf. [6, 19], that the Mabuchi functional is bounded from below (in the given class \(\alpha \)) if \(\alpha \) contains a cscK representative. In other words, Corollary 1.1 follows.
Moreover, it is shown in [8, Theorem 1.2] that the Mabuchi functional \(\mathrm {M}\) is in fact coercive if \(\alpha \) contains a cscK representative. As a consequence, we obtain also the following stronger result, confirming the “if” direction of the YTD conjecture (here referring to its natural generalisation to the transcendental setting, using the notions introduced in Sect. 3).
Corollary 5.5
If the Kähler class \(\alpha \in H^{1,1}(X,\mathbb {R})\) admits a constant scalar curvature representative, then \((X,\alpha )\) is uniformly Kstable.
5.3 Asymptotic Slope of the KEnergy
Adapting the techniques of [13] to the present setting we now obtain the following result, corresponding to Theorem C of the introduction.
Theorem 5.6
Remark 5.7
In particular, this result holds when \((\varphi _t)_{t \geqslant 0}\) is the weak geodesic ray associated with \((\mathcal {X},\mathcal {A})\), constructed in Sect. 4.1.
Proof of Theorem 5.6
Following ideas of [13] we associate with the given smooth, relatively Kähler and dominating test configuration \((\mathcal {X}, \mathcal {A})\) for \((X,\alpha )\) another test configuration \((\mathcal {X}', \mathcal {A}')\) for \((X,\alpha )\) which is semistable, i.e. smooth and such that \(\mathcal {X}'_0\) is a reduced \(\mathbb {R}\)divisor with simple normal crossings. As previously noted, we can also assume that \(\mathcal {X}'\) dominates the product. In the terminology of Sect. 3.6, this construction comes with a morphism \( g_d \circ \rho : \mathcal {X}' \rightarrow \mathcal {X}\), cf. the diagram in Section 3.6. Pulling back, we set \(\mathcal {A}' := g_d^*\rho ^*\mathcal {A}\). Note that \(\mathcal {A}'\) is no longer relatively Kähler, but merely relatively semipositive (with the loss of positivity occurring along \(\mathcal {X}_0'\)).
Lemma 5.8
Footnotes
 1.
We use the word ‘transcendental’ to emphasise that the Kähler class in question is not necessarily of the form \(c_1(L)\) for some ample line bundle L over X.
 2.
By a standard abuse of notation we identify a Hermitian metric h with its associated (1, 1)form \(\omega = \omega _h\) and refer to \(\omega \) as the “metric”.
 3.
The limit is in fact well defined, as shown in Sect. 5.3.
Notes
Acknowledgements
Since the first version of this paper was made available, R. Dervan and J. Ross, independently, used similar methods to establish Theorem A (see [25]). They are further able to establish Kstability of \((X,\omega )\) whenever the automorphism group \(\mathrm {Aut}(X,\omega )\) is discrete. We are grateful to them for helpful discussions on the topic of this paper and related questions. Moreover, it is a pleasure to thank Sébastien Boucksom, Vincent Guedj, Robert Berman and Ahmed Zeriahi, as well as the referees, for helpful discussions and suggestions.
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