From Monge–Ampère equations to envelopes and geodesic rays in the zero temperature limit
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Abstract
Let \((X,\theta )\) be a compact complex manifold X equipped with a smooth (but not necessarily positive) closed (1, 1)form \(\theta .\) By a wellknown envelope construction this data determines, in the case when the cohomology class \([\theta ]\) is pseudoeffective, a canonical \(\theta \)psh function \(u_{\theta }.\) When the class \([\theta ]\) is Kähler we introduce a family \(u_{\beta }\) of regularizations of \(u_{\theta }\), parametrized by a large positive number \(\beta ,\) where \(u_{\beta }\) is defined as the unique smooth solution of a complex Monge–Ampère equation of Aubin–Yau type. It is shown that, as \(\beta \rightarrow \infty ,\) the functions \(u_{\beta }\) converge to the envelope \(u_{\theta }\) uniformly on X in the Hölder space \(C^{1,\alpha }(X)\) for any \(\alpha \in ]0,1[\) (which is optimal in terms of Hölder exponents). A generalization of this result to the case of a nef and big cohomology class is also obtained and a weaker version of the result is obtained for big cohomology classes. The proofs of the convergence results do not assume any a priori regularity of \(u_{\theta }.\) Applications to the regularization of \(\omega \)psh functions and geodesic rays in the closure of the space of Kähler metrics are given. As briefly explained there is a statistical mechanical motivation for this regularization procedure, where \(\beta \) appears as the inverse temperature. This point of view also leads to an interpretation of \(u_{\beta }\) as a “transcendental” Bergman metric.
1 Introduction
In this paper we introduce a natural family of regularizations \(u_{\beta }\) of the envelope \(u_{\theta },\) indexed by a positive real parameter \(\beta ,\) where \(u_{\beta }\) is determined by an auxiliary choice of volume form dV; the functions \(u_{\beta }\) will be defined as solutions to certain complex Monge–Ampère equations, parametrized by \(\beta .\) Several motivations for studying the functions \(u_{\beta }\) and their asymptotics as \(\beta \rightarrow \infty ,\) will be given below. For the moment we just mention that \(u_{\beta }\) can, in a certain sense, be considered as a “transcendental” analog of the Bergman metric for a high power of a line bundle L over X and moreover from a statistical mechanical point of view the limit \(\beta \rightarrow 0\) appears as a zerotemperature limit.
In order to introduce the precise setting and the main results we start with the simplest case of a Kähler class\([\theta ].\) First note that the envelope construction above can be seen as a generalization of the process of replacing the graph of a given smooth functions with its convex hull. By this analogy it is already clear from the onedimensional case that \(u_{\theta }\) will almost never by \(C^{2}\)smooth even if the class \([\theta ]\) is Kähler (unless \(\theta \) is semipositive, so that \(u_{\theta }=0).\)
Theorem 1.1
Let \(\theta \) be a smooth (1, 1)form on a compact complex manifold X such that \([\theta ]\) is a Kähler class. Denote by \(u_{\theta }\) the corresponding \(\theta \)psh envelope and by \(u_{\beta }\) the unique smooth solution of the complex Monge–Ampère equations 1.1 determined by \(\theta \) and a fixed volume form dV on X. Then, as \(\beta \rightarrow \infty ,\) the functions \(u_{\beta }\) converge to \(u_{\theta }\) in \(\mathcal {C}^{1,\alpha }(X)\) for any \(\alpha \in ]0,1[,\) with a uniform bound on \(dd^{c}u_{\beta }.\)
More generally, we will consider the case when the cohomology class \([\theta ]\) is merely assumed to be big; this is the most general setting where complex Monge–Ampère equations of the form make sense [18]. The main new feature in this general setting is the presence of \(\infty \)singularities of all \(\theta \)psh functions on X. Such singularities are, in general, inevitable for cohomological reasons. Still, by the results in [18], the corresponding complex Monge–Ampère equations admit a unique \(\theta \)psh function \(u_{\beta }\) with minimal singularities; in particular its singularities can only appear along a certain complex subvariety of X, determined by the class \([\theta ]\), whose complement is called the Kähler locus\(\Omega \) of \([\theta ]\) (or the ample locus) introduced in [17] (which in the algebrogeometric setting corresponds to the complement of the augmented base locus of the corresponding line bundle). Moreover, in the case when the class \([\theta ]\) is also assumed to be nef the solution \(u_{\beta }\) is known to be smooth on \(\Omega ,\) as follows from the results in [18]. In this general setting our main result may be formulated as follows:
Theorem 1.2
Let \(\theta \) be a smooth (1, 1)form on a compact complex manifold X such that \([\theta ]\) is a big class. Then, as \(\beta \rightarrow \infty ,\) the functions \(u_{\beta }\) converge to \(u_{\theta }\) uniformly, in the sense that \(\left\ u_{\beta }u_{\theta }\right\ _{L^{\infty }Z(X)}\rightarrow 0.\) Moreover, if the class \([\theta ]\) is also assumed to be nef, then the convergence holds in \(\mathcal {C}_{loc}^{1,\alpha }(\Omega )\) on the Kähler locus \(\Omega \) of X.
1.1 Degenerations induced by a divisor and applications to geodesic rays
In the case of a Kähler class and when \(\theta \) is positive, i.e. \(\theta \) is Kähler form, it follows immediately from the definition that \(u_{\theta }=0\) and in this case the convergence in Theorem 1.1 holds in the \(\mathcal {C}^{\infty }\)sense, as recently shown in [36] using a completely different proof. However, as shown in [45, 47] in the integral case \([\omega ]=c_{1}(L),\) a nontrivial variant of the previous envelopes naturally appear in the geometric context of test configurations for the polarized manifold (X, L), i.e. \(\mathbb {C}^{*}\)equivariant polarized deformations \((\mathcal {X},\mathcal {L})\) of (X, L) and they can be used to construct (weak) geodesic rays in the space of all Kähler metrics in \([\omega ].\) Such test configurations were introduced by Donaldson in his algebrogeometric definition of Kstability of a polarized manifold (X, L), which according to the the Yau–Tian–Donaldson is equivalent to the existence of a Kähler metric in the class \(c_{1}(L)\) with constant scalar curvature. Briefly, Kstability of (X, L) amounts to the positivity of the Donaldson–Futaki invariants for all test configurations, which in turn is closely related to the large time asymptotics of Mabuchi’s Kenergy functional along the corresponding geodesic rays (see [42] and references therein).
Remarkably, as shown in [45, 47] (in the line bundle case) taking the Legendre transform of the envelopes \(u_{\lambda }+\lambda \log \left\ s\right\ ^{2}\) with respect to \(\lambda \) produces a geodesic ray in the closure of the space of Kähler potentials in \([\omega ],\) which coincides with the \(C^{1,\alpha }\)geodesic constructed by PhongSturm [40, 41] (in general, the geodesics are not \(C^{2}\)smooth). Here, building on [45, 47], we show that the logarithm of the Laplace transform, with respect to \(\lambda ,\) of the Monge–Ampère measures of the envelopes \(u_{\lambda }\) defines a family of subgeodesics in the space of Kähler potentials converging to the corresponding geodesic ray (see Corollary 5.3). In geometric terms the result may be formulated as follows
Corollary 1.3
This can be seen as a “transcendental” analogue of the approximation result of PhongSturm [44], which uses Bergman geodesic rays. However, while the latter convergence result holds pointwise almost everywhere and for t fixed, an important feature of the convergence in the previous corollary is that it is uniform, even when t ranges in all of \([0,\infty [.\) More generally, we will establish an extension of the previous result to the case when \([\omega ]c[Z]\) (or equivalently \(\mathcal {L}_{c})\) is merely assumed big.
The motivation for considering this “transcendental” approximation scheme for geodesic rays is twofold. First, as is wellknown, recent examples indicate that a more “transcendental” notion of Kstability is needed for the validity of the Yau–Tian–Donaldson conjecture, obtained by relaxing the notion of a test configuration. One such notion, called analytic test configurations, was introduced in [47] and as shown in op. cit. any such test configuration determines a weak geodesic ray, which a priori has very low regularity. However, the approximation scheme above could be used to regularize the latter weak geodesic rays, which opens the door for defining a notion of generalized Donaldson–Futaki invariant by studying the large time asymptotics of the Kenergy functional along the corresponding regularizations (as in the Bergman metrics approach in [44]). In another direction, the approximation scheme above should be useful when considering the analog of Kstability for a nonintegral Kähler class \([\omega ]\) (compare Sect. 5). The previous corollary is just a first illustration of this approximation scheme and we leave the development of more general approximation results for the future.
1.1.1 On the proofs
Next, let us briefly discuss the proofs of the previous theorems, starting with the case of a Kähler class. First, the weak convergence of \(u_{\beta }\) towards \(u_{\theta }\) (i.e. convergence in \(L^{1}(X))\) is proved using variational arguments (building on [12]). In fact, we will give two different proofs of this convergence, where the first one is variational and has two merits: (1) it generalizes directly to the case of a big class and (2) it applies when dV is replaced with a quite singular measure \(\mu _{0}\) (satifying a Bernstein–Markov property). The second proof uses a direct simple maximum principle argument.
In either way, to conclude the proof of Theorem 1.1 we just have to provide a priori estimates on \(u_{\beta },\) which are uniform in \(\beta \) and which we deduce from Siu’s variant of the Aubin–Yau Laplacian estimates. In particular, this implies convergence in \(L^{\infty }(X).\) However, in the case of a general big class, in order to establish the global \(L^{\infty }\)convergence, we need to take full advantage of the variational argument, namely that the argument shows that \(u_{\beta }\) converges to \(u_{\theta }\) in energy and not only in \(L^{1}(X).\) This allows us to invoke the \(L^{\infty }\)stability results in [32]. Briefly, the point is that convergence in energy implies convergence in capacity, which together with an \(L^{p}\)control on the corresponding Monge–Ampère measures opens the door for Kolodziej type \(L^{\infty }\)estimates. Moreover, a variant of the maximum principle argument used in the case of the Kähler class, based on the theory of viscosity subsolutions developed in [29], yields the bound 1.2 (only the local case of the results in [29] is needed).
1.2 Further background and motivation
Before turning to the proofs of the results introduced above it may be illuminating to place the result into a geometric and probabilistic context (see also Sect. 3.1 for the relation to Bergman kernel asymptotics).
1.2.1 Kähler–Einstein metrics and the continuity method
Interestingly, as observed in [50] the equation 1.3 can also be obtained from the Ricci flow via a backwards Euler discretization. Accordingly, the corresponding continuity path is called the Ricci continuity path in the recent paper [36], where it (or rather its “conical” generalization) plays a crucial role in the construction of Kähler–Einstein metrics with edge/cone singularities, by deforming the “trivial” solution \(\omega _{\beta }=\theta \) at \(\beta =\infty \) to a Kähler–Einstein metric at \(\beta =\pm 1.\) It should however be stressed that the main point of the present paper is to study the case of a nonpositive form \(\theta \) which is thus different from the usual settings appearing in the context of Kähler–Einstein geometry and where, as we show, the limit as \(\beta \rightarrow \infty \) is a canonical positive current associated to \(\theta .\)
1.2.2 Cooling down: the zero temperature limit
1.2.3 Added in proof
It has been pointed out by experts that the proof of the main result in [14], saying that the Laplacian of \(u_{\theta }\) is in \(L_{loc}^{\infty }\) on the Kähler locus of X, is incomplete (further details need to be added about how to obtain the estimate 1.8 in [14]). The authors intend to complement the proof given in [14] in the future, but in the case of a nef and big class the present paper provides a direct PDE proof of the regularity in question. In the case of a big, but nonnef class, the bound 1.2 is weaker than the bound in [14], but it appears to be adequate for all current complex geometric applications of envelopes as above, such as the recent proof of the duality between the pseudoeffective and the movable cone on a projective manifold in [62].
Since the first preprint version of the present paper appeared on ArXiv there has been a number of interesting developments that we briefly describe. In [24] it was shown that \(u_{\theta }\) is Lipschitz continuous as soon as \(\theta \) has a Lipschitz potential, using the regularizations \(u_{\beta }\) above and Blocki’s gradient estimate (as a replacement of the Aubin–Yau–Siu inequality used in Proposition 2.6). Moreover, very recently the convergence result for \(u_{\beta }\) in the present paper was used in [22, 54] to prove the \(C^{1,1}\)regularity of \(u_{\theta }\) (in the case of a Kähler class), by using the recent \(C^{1,1}\)estimates in [21] as a replacement of the Aubin–Yau–Siu inequality. In another direction it was shown in [38] how to extend the \(C^{0}\)convergence implicit in Theorem 1.1 to the setting of Hessian equations on Kähler manifolds, leading to a new global regularization result for \((\omega ,m)\)subharmonic functions (see Remark 3.5). Furthermore, very recently it was shown in [51] and [28], independently, that a transcendtal Kähler class containing a constant scalar curvature metric is Ksemistable, in general, and Kstable [28] if the automorphism group is discrete, which thus establishes one direction of the generalized Yau–Tian–Donaldson conjecture discussed in Sect. 5.0.2. Moreover, solutions \(u_{\beta }\) of global complex Monge–Ampère equations as above and their relative positivity properties were used in [20] to give an alternative proof of Chen’s conjecture concerning the convexity of the Kenergy (recently established in [9]) with \(u_{\beta }\) replacing the local Bergman metric approximations used in [9], which thus reinforces the intepretation of \(u_{\beta }\) as a transcendtal Bergman metric discussed in Sect. 3.1. See also the very recent work [33] for applications to viscosity theory. Finally, a dynamical analog of Theorem 1.1, formulated in terms of the zerotemperature limit of the twisted Kähler–Ricci flow, is obtained in [15].
1.2.4 Organization
After having setup the general framework in Sect. 2 we go on to first prove the main result (Theorem 1.1) in the case of Kähler class (by two different proofs) and then its generalization to big classes (Theorem 1.2). The interpretation in terms of transcendental Bergman metrics is discussed in Sect. 3, together with applications to regularization of \(\omega \)psh functions. Then in Sect. 4 we consider the singular version of the previous setup which appears in the presence of a divisor Z on X. Finally, the results in the latter section are applied in Sect. 5 to the construction and regularization of geodesic rays and relations to the transcendtal generalization of the Yau–Tian–Donaldson conjecture are discussed.
2 From Monge–Ampère equations to \(\theta \)psh envelopes
2.0.2 An alternative formulation in the Kähler case
We will be interested in the limit when \(\beta \rightarrow \infty .\) In order to separate the different kind of analytical difficulties which appear in the case when \([\theta ]\) is Kähler from those which appear in the general case when \([\theta ]\) is big, we will start with the Kähler case, even though it can be seen as a special case of the latter.
2.1 The case of a Kähler class (Proof of Theorem 1.1)
In this section we will assume that \([\theta ]\) is a Kähler class, i.e. there exists some smooth function \(v\in PSH(X,\theta )\) such that \(\omega :=\theta +dd^{c}v>0,\) i.e. \(\omega \) is a Kähler form.
2.1.1 Convergence in energy
Theorem 2.1
Let \(\mu _{0}\) be a finite measure on X not charging pluripolar subsets. Denote by \(u_{\beta }\) the solution to the complex Monge–Ampère equation determined by the data \((\theta ,\mu _{0},\beta ).\) If \(\mu _{0}\) has the Bernstein–Markov property wrt \(PSH(X,\theta ),\) then \(u_{\beta }\) converges to \(u_{\theta }\) in energy.
Proof
Step two: \(u_{\theta }\) is the unique supnormalized maximizer of \(\mathcal {G}_{\infty }\)
Finally, by the Bernstein–Markov property we have that \(\lim _{\beta \rightarrow \infty }\mathcal {L}_{\beta }(U_{\beta })=\lim _{\beta \rightarrow \infty } \sup (U_{\beta })=0\) and hence \(u_{\beta }\) also converges to \(u_{\theta }\) in \(L^{1}(X).\) Moreover, by Step one, we have \(\mathcal {E}(u_{\beta })\rightarrow \mathcal {E}(u_{\theta }),\) which concludes the proof of the theorem. \(\square \)
Remark 2.2
The present definition of the Bernstein–Markov property is the natural “transcendental” generalization of the definition used in [11, Definition 1.9], which concerns the case when \([\theta ]=c_{1}(L)\) for a big line bundle L. More generally, as in [11, Definition 1.9] one can consider the setting where a compact subset K of X has been fixed and say that a measure \(\mu _{0}\) supported on K has the Bernstein–Markov property wrt\(PSH(X,\theta )\)forK if the inequality 2.7 holds when X has been replaced with K. Repeating the proof in the previous theorem then shows that if the latter Bernstein–Markov property holds, then \(u_{\beta }\) converges to \(u_{\theta ,K}\) defined as in formula 2.1 (with X replaced by K) under the condition that \(u_{\theta ,K}\) be continuous (i.e. \((K,\theta )\) is regular in the sense of [11]).
In the case when \([\theta ]\) is a Kähler class we will only need the \(L^{1}\)convergence implicit in the previous theorem. But it should be stressed that when we move on to the case of a big class the convergence in energy will be crucial in order to establish the convergence in \(L^{\infty }\)norms.
2.1.2 A direct proof using the maximum principle when \(\mu _{0}\) is a volume form
Next we show how to give an alternative direct proof of the convergence of \(u_{\beta }\) towards \(u_{\theta }\) (in the case of a given volume form dV) by exploting that \(u_{\beta }\) is smooth, by the Aubin–Yau theorem. It gives a quantitative \(L^{\infty }\)convergence.
Proposition 2.3
Proof
2.1.3 \(L^{\infty }\)estimates
We start with the following wellknown
Lemma 2.4
Assume that u and v are (say, bounded) \(\theta \)psh functions such that \(MA(v)\ge e^{\beta v}dV\) and \(MA(u)\le e^{\beta u}dV.\) Then \(v\le u.\)
Proof
In the smooth case this follows immediately from the maximum principle and in the general case we can apply the comparison principle (which, by [18, Corollary 2.3], holds in the general setting of a big class considered below). Indeed, according to the comparison principle \(\int _{\{u\le v\}}MA(v)\le \int _{\{u\le v\}}MA(u)\) and hence \(\int _{\{u\le v\}}e^{\beta v}dV\le \int _{\{u\le v\}}e^{\beta u}dV.\) But then it must be that \(v\le u\) a.e. on X and hence everywhere. \(\square \)
The previous lemma allows us to construct “barriers” to show that \(u_{\beta }\) is uniformly bounded:
Lemma 2.5
Given \(\beta _{0}>0\) there exists a constant C such that \(\sup _{X}u_{\beta }\le C\) when \(\beta \ge \beta _{0}.\)
Proof
Take \(\beta \) such that \(\beta \ge \beta _{0}\) (the given positive number). Let us start with the proof of the lower bound on \(u_{\beta }.\) Since \([\theta ]\) is a Kähler class there is a smooth \(\theta \)psh function v such that \(MA(v)\ge e^{A}dV\) for some constant A. After shifting v by a constant we may assume that \(v\le A/\beta _{0}\le A/\beta .\) But then \(MA(v)\ge e^{A}dV\ge e^{\beta v}\) and hence by the previous lemma \(v\le u_{\beta }\) which concludes the proof of the lower bound. Similarly, taking v to be a smooth \(\theta \)psh function v such that \(MA(v)\le e^{A}dV\) and shifting v so that \(A/\beta _{0}\le v\) proves that \(u_{\beta }\le v,\) which concludes the proof of the lemma. \(\square \)
2.1.4 The Laplacian estimate
Next we will establish the following key Laplacian estimate:
Proposition 2.6
Proof
Remark 2.7
Note that, in general, the Ricci curvature of the Kähler forms \(\omega _{\beta }:=\theta +dd^{c}u_{\beta }\) is unbounded, both from above and below, as \(\beta \rightarrow \infty .\) Still, by the previous estimate, the Kähler forms \(\omega _{\beta }\) are uniformly bounded from above. However it should be stressed that, unless \(\theta >0,\) there is no uniform bound of the form \(\omega _{\beta }\ge \delta \omega >0\) as it will follow from Theorem 1.1 that \(\omega _{\beta }^{n}\rightarrow 0\) on large portions of X (indeed, for \(\beta \) large, \(\omega _{\beta }^{n}\le Ce^{\beta \epsilon }dV\) on the open set where \(u_{\theta }<2\epsilon ).\)
2.1.5 Proof of Theorem 1.1 using the variational approach
By Lemma 2.5\(u_{\beta }\) is uniformly bounded and by the Laplacian estimate in Proposition 2.6 combined with Green’s formula the gradients of \(u_{\beta }\) are uniformly bounded. Hence, it follows from basic compactness results that, after perhaps passing to a subsequence, \(u_{\beta }\) converges to a function u in \(\mathcal {C}^{1,\alpha }(X)\) for any fixed \(\alpha \in ]0,1[.\) It will thus be enough to show that \(u=u_{\theta }\) (since this will show that any limit point of \(\{u_{\beta }\}\) is uniquely determined and coincides with \(u_{\theta }\)). But this follows from either Theorem 2.1 or Proposition 2.3.
2.2 The case of a big class (proof of Theorem 1.2)
A (Bott–Chern) cohomology class \([\theta ]\) in \(H^{1,1}(X)\) is said to be big, if \([\theta ]\) contains a Kähler current \(\omega ,\) i.e. a positive current \(\omega \) such that that \(\omega \ge \epsilon \omega _{0}\) for some positive number \(\epsilon ,\) where \(\omega _{0}\) is a fixed strictly positive form \(\omega _{0}\) on X. We also recall that a class \([\theta ]\) is said to be nef if, for any \(\epsilon >0,\) there exists a smooth form \(\omega _{\epsilon }\in T\) such that \(\omega _{\epsilon }\ge \epsilon \omega _{0}.\) To simplify the exposition we will assume that X is a Kähler manifold so that the form \(\omega _{0}\) may be chosen to be closed. Then the cone of all big classes in the cohomology group \(H^{1,1}(X)\) may be defined as the interior of the cone of pseudoeffective classes and the cone of Kähler classes may be defined as the interior of the cone of nef classes.
We also recall that a function u in \(PSH(X,\theta )\) is said to have minimal singularities, if for any \(v\in PSH(X,\theta )\) the function \(uv\) is bounded from below on X. In particular, the envelope \(u_{\theta }\) has (by its very definition) minimal singularities (and this is in fact the standard construction of a function with minimal singularities). In the case when \([\theta ]\) is big any function with minimal singularities is locally bounded on a Zariski open subset \(\Omega ,\) as a wellknown consequence of Demailly’s approximation results [26]. In fact, the subset \(\Omega \) can be taken as the Kähler (ample) locus of \([\theta ]\) defined in [17].
Example 2.8
Let Y be a singular algebraic variety in complex projective space \(\P ^{N}\) and \(\omega \) a Kähler form on \(\P ^{n}\) (for example, \(\omega \) could be taken as the FubiniStudy metric so that \([\omega _{Y}]\) is the first Chern class of \(\mathcal {O}_{X}(1)\)). If now \(X\rightarrow Y\) is a smooth resolution of Y, which can be taken to invertible over the regular locus of Y; then the pullback of \(\omega \) to X defines a class which is nef and big and such that its Kähler locus corresponds to the regular part of Y.
We will denote by MA the Monge–Ampère operator on \(PSH(X,\theta )\) defined by replacing wedge products of smooth forms with the nonpluripolar product of positive currents introduced in [18]. The corresponding operator MA is usually referred to as the nonpluripolar Monge–Ampère operator. For example, if u has minimal singularities, then \(MA(u)=1_{\Omega }MA(u_{\Omega })\) on the Kähler locus \(\Omega ,\) where \(MA(u_{\Omega })\) may be computed locally using the classical definition of Bedford–Taylor. We let V stand for the volume of the class \([\theta ],\) which may be defined as the total mass of MA(u) for any function u in \(PSH(X,\theta )\) with minimal singularities. By [18] there exists a unique solution \(u_{\beta }\) to the Eq. 2.2 in \(PSH(X,\theta )\) with minimal singularities. Moreover, by [18] the solution is smooth on the Kähler locus in the case when \([\theta ]\) is nef and big (which is expected to be true also without the nef assumption; compare the discussion in [18]).
2.2.1 Convergence in energy
Remark 2.9
Strictly speaking, in the case of a Kähler class the definition 2.13 of \(\mathcal {E}\) only coincides with the previous one (formula 2.5) in the case when \(\theta \) is semipositive (since the definition in formula 2.5 corresponds to the normalization condition \(\mathcal {E}(0)=0).\) But the point is that, in the Kähler case, different normalizations gives rise to functionals which only differ up to an overall additive constant and hence the choice of normalization does not effect the notion of convergence in energy.
The proof of Theorem 1.1 can now be repeated word for word to give the following
Proposition 2.10
Suppose that \(\theta \) is a smooth form such that the class \([\theta ]\) is big. Then \(u_{\beta }\) converges to \(u_{\theta }\) in energy.
2.2.2 \(L^{\infty }\)estimates
We will also need the following upper bound on \(u_{\beta }:\)
Lemma 2.11
Proof
We recall that in the case of a Kähler class the estimate in the previous lemma was obtained as consequence of the maximum principle in the proof of Proposition 2.3. Next, we generalize the \(L^{\infty }\)convergence in Proposition 2.3 to a general big class, using the convergence in energy in Proposition 2.10.
Proposition 2.12
Proof
2.2.3 Bound on the Monge–Ampère measure of \(u_{\theta }\)
2.2.4 Laplacian estimates
For the Laplacian estimate we will have to assume that the big class \([\theta ]\) is nef.
Proposition 2.13
Suppose that the class \([\theta ]\) is nef and big. Then the Laplacian of \(u_{\beta }\) is locally bounded wrt \(\beta \) on the Zariski open set \(\Omega \subset X\) defined as the Kähler locus of X.
Proof
In the special case when \(\theta \) is semipositive and big (the latter condition then simply means that \(V>0)\) it follows from the results in [29] that \(u_{\beta }\) is continuous on all of X and hence Proposition 2.12 then says that \(u_{\beta }\rightarrow u_{\theta }\) in \(C^{0}(X).\)
Remark 2.14
2.2.5 End of the proof of Theorem 1.2 in the big case
This is proved exactly as in the case of a Kähler class, given the convergence results established above.
3 Transcendental Bergman metric asymptotics and Applications to regularization of \(\omega \)psh functions
3.1 Transcendental Bergman kernels
The main virtue of the family \(u_{\beta }\) is that it is canonically determined by the pair \((\theta ,dV))\) and exists also in the general transcendental setting of a Kähler class \([\theta ]\) which can not be realized as the first Chern class \(c_{1}(L)\) of a line bundle. Accordingly, it seems natural to expect that it can be used as a substitute for the timehonoured technique in complex geometry of using Bergman kernels as an approximation tool. In Sects. 3.2 and 5 we will give two such applications to the regularization problem of \(\omega \)psh functions and weak geodesic rays, respectively.
In the following it will be convenient to use the equivalent formulation of envelopes of the form \(P_{\omega }(f)\) in Sect. 2.0.2 (occasionally dropping the subscript \(\omega ).\) In other words, we start with a reference Kähler form \(\omega \) on X. Given a smooth function f we denote by \(P_{\beta }(f)\) the solution \(\varphi _{\beta }\) of the corresponding Monge–Ampère equation 2.4. In the line bundle setting above this corresponds to fixing a reference metric \(\left\ \cdot \right\ _{0}\) on L and writing \(\left\ \cdot \right\ ^{2}=\left\ \cdot \right\ _{0}e^{f}\) wich has curvature form \(\theta =\omega +dd^{c}f.\)
Lemma 3.1
Proof
The decreasing property follows directly from the comparison principle (Lemma 2.4) and the scaling property from the very definitions of \(P_{\beta }.\)\(\square \)
Remark 3.2
Let us briefly explain how the setting above fits into the statistical mechanical setup recalled in Sect. 1.2. The point is that one can let the inverse temperature \(\beta ,\) defining the probability measures 1.4, depend on k. In particular, for \(\beta =k\) one obtains a determinantal random point process. A direct calculation (compare [4]) reveals that the corresponding one point correlation measure \(\int _{X^{N_{k1}}}\mu _{k,\beta }\) then coincides with the Bergman measure \(\nu _{k}\) defined above. This means that the limit \(k\rightarrow \infty \) which appears in the “Bergman setting” can—from a statistical mechanical point of view—be seen as a limit where the number \(N_{k}\) of particles and the inverse temperature \(\beta \) jointly tend to infinity.
3.2 Regularization of \(\omega \)psh functions
In this section we consider the case of a Kähler class \([\omega ].\) We show how to give a simple global PDE proof of the following special case of the general regularization results of Demailly [26]:
Theorem 3.3
Let \([\omega ]\) be a Kähler class. Then any function \(\psi \in PSH(X,\omega )\) can be written as a decreasing limit of functions \(\psi _{j}\) which are smooth and strictly \(\omega \)psh.
Proof
It should be pointed out that by a local gluing argument of Richberg [43] the regularization result above can be reduced to the case of a continuous\(\omega \)psh function \(\psi \) (using the usual local regularizations involving convolutions). In turn, it was shown in [16] that the continuity assumption can be replaced by the assumption of vanishing Lelong numbers and hence, as explained in [16], approximating a general element \(\psi \in PSH(X,\omega )\) with the decreasing sequence \(\psi _{l}:=\max \{\psi ,l\}\) in \(PSH(X,\omega )\cap L^{\infty }\) gives a simple elemenary proof of the previous theorem. In the light of the discussion in the previous section the present global regularization scheme can be seen as a transcental analog of the wellknown Bergman kernel approach to regularization used in the line bundle setting (see [26, 31]). The present approach has the virtue of preserving higher order regularity properties of \(\psi \) as summarized in the following.
Theorem 3.4
Proof
Since \((\omega +dd^{c}f)^{n}\) has an \(L^{\infty }\)density [37] gives that \(\varphi \) is in \(C^{\alpha }(X)\) for some Hölder exponent \(\alpha '>0.\) By the complex generalization of EvansKrylov theory in [60] it then follows that \(\varphi _{\beta }\) is in \(C^{2,\alpha }(X)\) for some \(\alpha >0.\) Moreover, if \((\omega +dd^{c}\varphi )\) has coefficents in \(L^{\infty }\) then elliptic boot strapping gives that \(\varphi _{\beta }\) is in \(C^{3,\alpha }\) for any \(\alpha <1\) and Proposition 2.6 shows that \(\omega +dd^{c}\varphi _{\beta }\le C'\omega .\)\(\square \)
In particular, the transcendtal Bergman measure \(e^{k(P_{\beta }\varphi \varphi )}dV\) is uniformly bounded from above as long as \((\omega +dd^{c}\varphi )^{n}\) has an \(L^{\infty }\)density. For the ordinary Bergman measure the corresponding uniform bound was recently established in [9], under the stronger assumption that \((\omega +dd^{c}\varphi )\) has coefficents in \(L^{\infty }.\) The latter result was used in the proof, involving local Bergman metric approximations, of Chen’s conjecture concerning the convexity of the Kenergy along weak geodesics in the closure of the space of Kähler metrics.
Remark 3.5
Inspired by the first preprint version of the present paper on ArXiv it was shown in [38] how to use a genaralization of the transcental Bergman kernels introduced here, using Hessian equations as a substitute for Monge–Ampère equations, in order to establish the corresponding conjectural global regularization result for \((\omega ,m)\)subharmonic functions (i.e. usc functions u such that \((\omega +dd^{c}u)^{p}\wedge \omega ^{np}\ge 0\) for \(p=1,2\ldots ,m;\) the case \(m=n\) corresonds to the present setting). The elegant argument in [38] uses the notion of viscocity solutions of Hessian equations based on the technique introduced in [29].
4 Degenerations induced by a divisor
Note that it follows immediately from the definition that \(u_{\lambda }\) has minimal singularities. In particular, if \(\lambda <\epsilon ,\) then \(u_{\lambda }\) is bounded. In fact, \(u_{\lambda }\) is even continuous. The point is that, as long as the function \(\varphi _{0}\) is lower semicontinuous the corresponding envelope \(P_{\theta }(\varphi _{0})\) will also be continuous. Indeed, it follows immediately that \(P_{\theta }(\varphi _{0})^{*}\le \varphi _{0}\) and hence \(P_{\theta }(\varphi _{0})^{*}=P_{\theta }(\varphi _{0}),\) showing uppersemi continuity. The lower semicontinuity is then a standard consequence of Demailly’s approximation theorem applied to the Kähler class \([\theta ]\) (Theorem 3.3).
Theorem 4.1
Proof
Set \(f:=\left\ s\right\ ^{2},\) which is a lsc function \(X\rightarrow ]\infty ,\infty ]\) such that \(dd^{c}f\le C\omega .\) The convergence in energy and hence the uniforme convergence then follows as before. Finally, the uniform bound on \(dd^{c}u_{\beta ,\lambda }\) is obtained by writing f is a decreasing limit of smooth function \(f_{j}\) such that \(dd^{c}f_{j}\le C'\omega ,\) applying Proposition cr for a fixed j and finally letting \(j\rightarrow \infty .\)\(\square \)
Remark 4.2
More generally, it is enough to assume that \(\omega \) is semipositive and big; then the uniform bound on \(dd^{c}u_{\beta ,\lambda }\) in the previous theorem holds on any compact subset of the Kähler locus of X (by Proposition 2.13). For example, this situation appears naturally when Z is the expectional divisor in the blowup of a point on a Kähler manifold \((M,\omega _{M})\) and \(\omega \) is the pullback of M. Then the corresponding constant \(\epsilon \) is the Seshadri constant of p wrt \([\omega _{M}].\)
5 Applications to geodesic rays and test configurations

A normal variety \(\mathcal {X}\) with a \(\mathbb {C}^{*}\)action and flat equivariant map \(\pi :\mathcal {X}\rightarrow \mathbb {C}\)

A relatively ample \(\mathbb {Q}\)line bundle \(\mathcal {L}\) over \(\mathcal {X}\) equipped with an equivariant lift \(\rho \) of the \(\mathbb {C}^{*}\)action on X

An isomorphism of (X, L) with \((\mathcal {X},\mathcal {L})\) over \(1\in \mathbb {C}\)
Definition 1

A normal Kähler space \(\mathcal {X}\) equipped with a holomorphic \(S^{1}\)action and a flat holomorphic map \(\pi :\mathcal {X}\rightarrow \mathbb {C}.\)

An \(S^{1}\)equivariant embedding of \(X\times \mathbb {C}^{*}\) in \(\mathcal {X}\) such that \(\pi \) commutes with projection onto the second factor of \(X\times \mathbb {C}^{*}.\)

A (1, 1)cohomology Kähler class \([\Omega ]\) on \(\mathcal {X}\) whose restriction to \(X\times \{1\}\) may be identified with \([\omega ]\) under the previous embedding.
In particular, a test configuration \((\mathcal {X},\mathcal {L})\) for a polarized variety (X, L) induces a test configuration for \((X,c_{1}(L)).\) The point is that the \(\mathbb {C}^{*}\)action on \((\mathcal {X},\mathcal {L})\) induces the required isomorphism between \(\mathcal {X}\) and \(X\times \mathbb {C}^{*}\) over \(\mathbb {C}^{*}.\)
Proposition 5.1
Proof
Example 5.2
(deformation to the normal cone; compare [48, 49]). Any given (say reduced) divisor Z in X determines a special test configuration whose total space \(\mathcal {X}\) is the deformation to the normal cone of Z. In other words, \(\mathcal {X}\) is the blowup of \(X\times \mathbb {C}\) along the subscheme \(Z\times \{0\}.\) Denote by \(\pi \) the corresponding flat morphism \(\mathcal {X}\rightarrow \mathbb {C}\) which factors through the blowdown map p from \(\mathcal {X}\) to \(X\times \mathbb {C}.\) This construction also induces a natural embedding of \(X\times \mathbb {C}^{*}\) in \(\mathcal {X}.\) Given a Kähler class \([\omega ]\) on X, which we may identify with a class on \(X\times \mathbb {C}\) and a positive number c we denote by \([\Omega _{c}]\) the corresponding class \([p^{*}\omega ]c[E]\) on \(\mathcal {X},\) where E is the exceptional divisor and we are assuming that \(c<\epsilon ,\) where \(\epsilon \) is defined as the sup over all positive numbers c such that the class \([\Omega _{c}]\) is Kähler (i.e. \(\epsilon \) is the Seshadri constant of Z wrt \([\omega ]).\) In this setting it is not hard to check that \(\varphi \in \mathcal {F}^{\mu }(X,\omega )\) iff \(\nu _{Z}(\varphi )\ge \mu +c,\) where \(\nu _{Z}(\varphi )\) denotes the Lelong number of \(\varphi \) along the divisor Z in X. The point is that \([p^{*}\omega ]cE\) may be identified with the subspace of currents in \([p^{*}\omega ]\) with Lelong number at least c along the divisor E in \(\mathcal {X}\) which in this case is equivalent to having Lelong number at least c along the central fiber \([\mathcal {X}_{0}],\) which in turn is equivalent to \(\varphi \) having Lelong number at least c along Z in X. In particular, setting \(\mu =\lambda c\) we have \(\varphi _{\lambda }=\psi _{\mu },\) where \(\varphi _{\lambda }\) is the envelope defined by formula 4.1, i.e. \(u_{\lambda }=\psi _{\mu }\lambda \log \left\ s\right\ ^{2},\) where \(u_{\lambda }\) is defined by 4.2.
Theorem 5.3
Let \([\omega ]\) be a Kähler class on X and Z a divisor in X and fix a positive number \(c\in [0,\epsilon [.\) Then the corresponding subgeodesics \(\varphi _{\beta }^{t}\) converge, as \(\beta \rightarrow \infty ,\) to the weak geodesic \(\varphi ^{t},\) uniformly on \(X\times [0,T[\) for any fixed \(T<\infty \) (and for \(T=\infty \) in the case when \([\omega ]\in H^{2}(X,\mathbb {Q})).\) Moreover, the first order spacetime derivatives of \(\varphi _{\beta }^{t}\) are uniformly bounded on \(X\times [0,\infty ].\)
Proof
Remark 5.4
In the case when \([\omega ]=c_{1}(L)\) it was shown in [44] how to approximate (in a pointwise almost everywhere sense) a weak geodesic \(\varphi _{t}\) associated to a test configuration by smooth Bergman geodesics associated to higher powers of the line bundle L (see also [47] for an alternative proof). Accordingly, it seems natural to view \(\varphi _{\beta }^{t}\) as a transcendtal analog of the PhongSturm Bergman geodesics. One advantage of \(\varphi _{\beta }^{t}\) is that the convergence is uniform (even when t is not constrained to be in a bounded interval in the case of a rational class). Assuming the conjectural validity of the appoximation result in [23] for general transcendental classes, the uniformity in the previous theorem holds for \(T=\infty ,\) in general. It is also interesting to compare the bound on the first derivatives above with the case of toric Bergman geodesics studied in [58], where uniform \(C^{1}\)convergence is established. It seems likely that a similar \(C^{1}\)convergence holds in the present setting (even in the general nontoric setting), but we will not go further into this here. It would also be interesting to see if there is a uniform bound on the space Laplacians of \(\varphi _{\beta }^{t}\) (say on any fixed time inverval).
5.0.1. General (analytic) test configurations
Of course, the test configurations defined by the deformation to the normal cone of a divisor are very special ones. But the convergence result in Corollary 5.3 can be extended to general test configurations for a polarized manifold (X, L) (by replacing \(MA(u_{\beta ,\lambda })\) with \(MA(\varphi _{\beta ,\mu })\) where \(\varphi _{\beta ,\mu }\in \mathcal {F}^{\lambda }(X,\omega )\) satisfies the Eq. 4.3). The argument uses Odaka’s generalization of the RossThomas slope theory [39] defined in terms of a flag of ideals on X. The point is that by blowing up the corresponding ideals one sees that the pullback of the corresponding envelopes \(\psi ^{\mu }\) have divisorial singularities (compare Proposition 3.22 in [35]) so that the previous convergence argument can be repeated (as they apply also when L is merely semiample and big, which is the case on the blowup).
More generally, an analytic generalization of test configurations for a polarization (X, L) was introduced in [47]. Similarly, an analytic test configuration for a Kähler manifold \((X,\omega )\) may be defined as a concave family \([\psi ^{\mu }]\) of singularity classes in \(PSH(X,\omega ).\) The corresponding space \(\mathcal {F}^{\mu }(X,\omega )\) may then be defined as all elements \(\psi \) in such that \([\psi ]=[\psi ^{\mu }].\) To any such family one associates a family of envelopes \(\psi _{\mu }\) defined by formula 5.4. As shown in [47] taking the Legendre transform of \(\psi _{\mu }\) wrt \(\mu \) gives a curve \(\varphi ^{t}\) in \(PSH(X,\omega )\) which is a weak geodesic. The regularization scheme introduced in this paper could be adapted to this general framework by first introducing suitable algebraic regularizations of the singularity classes and using blowups (as in [39]). But we leave these developments and their relation to Kstability and the Yau–Tian–Donaldson conjecture for the future. For the moment we just observe that the latter conjecture admits a natural generalization to transcendental classes.
Example 5.5
Continuing with the previous example of deformation to the normal cone, we observe that one obtains a (transcendtal) analytic test configuration, which is not a bona fide test configuration, when \(c\in ]\epsilon ,\epsilon '[.\) In geometric terms this corresponds to allowing the line bundle \(\mathcal {L}\) (or the corresponding Kähler class on the total space) to be merely big. In this setting the \(C^{0}\)convergence in Theorem 5.3 still holds (with the same proof) as long as t is restricted to a bounded interval.
5.0.2. A generalization of the Yau–Tian–Donaldson conjecture to transcendetal classes
Footnotes
Notes
Acknowledgements
It is a pleasure to thanks David WittNyström for illuminating discussions on the works [47, 61], Julius Ross for inspiring discussions on the Yau–Tian–Donaldson conjecture for transcendantal Kähler classes, JeanPierre Demailly for the stimulating colaboration [14], which was one of the motivations for the current work and Chinh Lu and Yanir Rubinstein for helpful comments on the first preprint version of the present paper. Also thanks to the referee for many helpful comments. This work was partly supported by grants from the European Research Council and Knut and Alice Wallenberg foundation.
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