Kähler geometry of horosymmetric varieties, and application to Mabuchi’s K-energy functional

Thibaut Delcroix

doi:10.1515/crelle-2018-0040

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Kähler geometry of horosymmetric varieties, and application to Mabuchi’s K-energy functional

Thibaut Delcroix

Published/Copyright: January 20, 2019

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2020 Issue 763

Abstract

We introduce a class of almost homogeneous varieties contained in the class of spherical varieties and containing horospherical varieties as well as complete symmetric varieties. We develop Kähler geometry on these varieties, with applications to canonical metrics in mind, as a generalization of the Guillemin–Abreu–Donaldson geometry of toric varieties. Namely we associate convex functions with Hermitian metrics on line bundles, and express the curvature form in terms of this function, as well as the corresponding Monge–Ampère volume form and scalar curvature. We provide an expression for the Mabuchi functional and derive as an application a combinatorial sufficient condition of properness similar to one obtained by Li, Zhou and Zhu on group compactifications. This finally translates to a sufficient criterion of existence of constant scalar curvature Kähler metrics thanks to the recent work of Chen and Cheng. It yields infinitely many new examples of explicit Kähler classes admitting cscK metrics.

A Integration away from the walls

For this appendix we work on a finite-dimensional Euclidean vector space (V,〈⋅,⋅〉). Given R⊂V, s≤t∈ℝ∩{±∞}, we set

Σ⁢(R,s,t)={x:s≤〈α,x〉≤t⁢ for all ⁢α∈R}.

Let S denote a finite set of unit vectors in V such that each α∈R is the interior pointing unit normal vector to a facet of the cone Σ⁢(S,0,∞) (in particular, we implicitly assume that the cone is of the same dimension as V). Let Δ denote a convex body contained in the cone Σ⁢(S,0,∞). We introduce two assumptions, to be used in later statements. The first one concerns Δ and S:

Assumption A.1.

The convex body Δ satisfies Δ∩σ⁢(S,0,0)≠∅ and there exists ϵ>0 such that for any α∈S,

Δ∩Σ⁢({α},0,ϵ)=((Δ∩α⊥)×[0,ϵ]⁢α)∩Σ⁢(S,0,∞)

(note that the decomposition on the right-hand side is with respect to the orthogonal decomposition V=α⊥⊕ℝ⁢α).

The second assumption is an assumption on functions w:Δ→ℝ:

Assumption A.2.

The function w:Δ→ℝ is non-negative, convex and for any subset R⊂S, for any p∈Σ⁢(R,0,0), the directional derivative of w at p is non-negative for any direction ξ∈Σ∩Vect⁢(R).

Let dp denote a Lebesgue measure on V. The goal of this appendix is to provide a proof to the following general version of [35, Lemma 4.6]. Apart from presentation, the proof is identical to the original, we include it as a courtesy to the reader and since the version we use in the core of the article does not follow from the statement of [35, Lemma 4.6].

Proposition A.3.

Assume that (S,Δ) satisfy Assumption A.1. Then there exist ϵ>0 and C>0 such that for any w:Δ→R satisfying Assumption A.2,

∫Δw⁢𝑑p≤C⁢∫Δ∩Σ⁢(S,ϵ,∞)w⁢𝑑p.

Note that the two assumptions are obviously satisfied if S is the set of simple roots of a restricted root system, Δ is the polytope associated to a polarized horosymmetric variety satisfying assumption (T) as in Section 7 and w is the restriction of the convex conjugate of a normalized toric potential (which is invariant under the restricted Weyl group hence satisfy Assumption A.2). The statement used in the core of the article is rather the following direct consequence, applied to the Duistermaat–Heckman polynomial.

Corollary A.4.

Assume that (S,Δ) satisfy Assumption A.1, and assume that g:Δ→R is such that for any ϵ>0, infΔ∩Σ⁢(S,ϵ,∞)⁡g>0. Then there exists C>0 such that for any w:Δ→R satisfying Assumption A.2,

∫Δw⁢𝑑p≤C⁢∫Δw⁢g⁢𝑑p.

Proof of Proposition A.3.

The proof goes by (strong) induction on the cardinality of S and relies on two ingredients. The first and main ingredient is a decomposition of Δ into several parts using Assumption A.1 such that the induction hypothesis applies to all but one. The second ingredient is an elementary use of convexity and Assumption A.2 to deal with the remaining part.

Note already that the initialization is trivial: if S is empty, then the statement is also empty. We now describe the decomposition of Δ. Let 𝒮 denote the set of all subsets R of S satisfying Δ∩σ⁢(R,0,0)≠∅. It contains S itself by assumption. Fix an ϵ>0 small enough so that Assumption A.1 holds with this value of ϵ. Set ΔS:=Δ∩Σ⁢(S,0,ϵ2): then in the orthogonal decomposition V=Σ⁢(S,0,0)⊕Vect⁢(S), Assumption A.1 shows that

ΔS=(Δ∩Σ⁢(S,0,0))×(Σ⁢(S,0,ϵ2)∩Vect⁢(S)).

We can then define, by induction, the subsets ΔR for R∈𝒮 by setting

ΔR:=(Δ∖⋃R⊂R′ΔR′)∩Σ⁢(R,0,ϵ2).

Each (R,ΔR) satisfies Assumption A.1 and the cardinality of R is smaller than that of S when R≠S, so by induction hypothesis we may find an ϵ0>0 and C0>0 such that

∫Δ∖ΔSu⁢𝑑p≤C0⁢∫(Δ∖ΔS)∩Σ⁢(S,ϵ0,∞)u⁢𝑑p.

Finally, we treat the case of ΔS. Let F denote the set

F={x∈Vect⁢(S)∩Σ⁢(S,0,ϵ):there exists an α∈S such that ⁢〈α,x〉=ϵ}.

Then we have

∫ΔSw⁢𝑑p=∫Δ∩Σ⁢(S,0,0)∫Σ⁢(S,0,ϵ2)∩Vect⁢(S)w⁢(x,y)⁢𝑑y⁢𝑑x

=∫Δ∩Σ⁢(S,0,0)∫012∫Fw⁢(t⁢f,y)⁢𝑑σ⁢𝑑t⁢𝑑y,

where d⁢σ denotes the area measure. By Assumption A.2, we obtain for the last expression that

∫Δ∩Σ⁢(S,0,0)∫012∫Fw⁢(t⁢f,y)⁢𝑑σ⁢𝑑t⁢𝑑y≤∫Δ∩Σ⁢(S,0,0)∫121∫Fw⁢(t⁢f,y)⁢𝑑σ⁢𝑑t⁢𝑑y

≤∫Δ∅w⁢𝑑p.

This finishes the proof. ∎

B Properness on invariant potentials and existence of constant scalar curvature metrics

Let X be a complex projective manifold, and let be L an ample line bundle on X. Assume that (X,L) is equipped with two actions:

the action of a compact Lie group acting K,
the action of a connected real Lie group N which normalizes the action of K, that is, for any n∈N, k∈K, there exists k′∈K such that k⋅(n⋅x)=n⋅(k′⋅x).

Recall that we defined in Sections 7.3.1 and 7.3.2 the functionals J and Mab (in this appendix, Θ is empty) on the space of smooth and invariant Kähler potentials rPSHK⁢(X,ωref) for a fixed reference metric ωref∈c1⁢(L). Recall that the values of Mab and J depend only on the Kähler metric ωϕ=ωref+i⁢∂⁡∂¯⁢ϕ defined by ϕ. In particular, it makes sense to define J⁢(n⋅ϕi) for n∈N without fixing a normalization of the potentials.

Definition B.1.

The functional Mab is proper modulo N on smooth K-invariant potentials if

it is bounded from below on smooth K-invariant potentials, and
any sequence (ϕi) of smooth K-invariant potentials such that infn∈N⁡J⁢(n⋅ϕi)→∞ must satisfy Mab⁢(ϕi)→∞.

A few remarks are in order before moving on:

Properness with respect to J, as defined here, is equivalent to properness with respect to Mabuchi’s L1 distance d1 in restriction to potentials normalized by vanishing of the Aubin-Mabuchi functional thanks to [16, Proposition 5.5].
It is standard that boundedness from below of the Mabuchi functional on smooth K-invariant potentials implies N-invariance of the Mabuchi functional by linearity on the families of metrics induced by real one-parameter subgroups of the connected group N (see e.g. [15, Lemma 3.3], the restriction to invariant potentials makes no difference here).
Our definition of coercivity (Definition 7.9) implies properness modulo Z⁢(G)0 on smooth K-invariant potentials of the Mabuchi functional in the setting of the article.

A close examination of Chen and Cheng’s arguments in [15] allows to obtain the following statement.

Theorem B.2.

Assume that the Mabuchi functional Mab is proper modulo N on smooth K-invariant Kähler potentials, then there exists a constant scalar curvature metric in c1⁢(L).

Proof.

Starting from a K-invariant reference metric ω0 (which exists by averaging), we consider the continuity path of twisted constant scalar curvature equations: for t∈[0,1],

t⁢(Sϕ-S¯)=(1-t)⁢(trωϕ⁡ω0-n).

By [14, Corollary 4.5] and [5, Theorem 4.7], a solution to the above equation is unique as long as t<1. Since ω0 is K-invariant, it implies that any solution for t<1 is K-invariant. This simple remark allows to apply [14, Theorem 4.1] and [15, Lemma 3.6] in restriction to invariant potentials to obtain solutions for any t<1.

Let now ti denote a sequence of elements of [0,1] increasing to 1. Let ϕ~i denote the corresponding solutions. By [15, Lemma 3.7], the Mabuchi functional is uniformly bounded along the solutions ϕ~i. Properness modulo N on invariant potentials then implies

supi⁡infn∈N⁡J⁢(n⋅ϕ~i)<∞.

The conclusion of the theorem then follows directly from [15, Proposition 3.9]. ∎

C Properness on invariant potentials and existence of log-Kähler–Einstein metrics

Let (X,Θ) denote a log-Fano klt pair, and assume that X is smooth, for simplicity in dealing with smooth Kähler metrics. Assume as in the previous appendix that X is equipped with two actions, both stabilizing each component of Θ:

the action of a compact Lie group acting K,
the action of a connected real Lie group N which normalizes the action of K, that is, for any n∈N, k∈K, there exists k′∈K such that k⋅(n⋅x)=n⋅(k′⋅x).

In order to follow more closely [4], we introduce some notations closer to theirs. We fix a reference metric ωref in c1⁢(X)-Θ. Let ℰ1 denote the space of finite energy potentials with respect to ωref as defined in [4, Section 1.4]. It maps bijectively to the space 𝒯1 of finite energy currents via the map ϕ↦ωϕ=ωref+i⁢∂⁡∂¯⁢ϕ and 𝒯1 maps to the space ℳ1 of finite energy probability measures via the map ω↦V-1⁢ωn, where V denotes the volume of (X,L). We further let ϕ↦MA⁢(ϕ)=V-1⁢ωϕn denote the composition of the two maps, and let ω↦ϕω denote the inverse map from 𝒯1 to ℰ1.

Let μref denote the adapted measure of the pair (X,Θ) (see [4, Definition 3.1]). Another finite energy probability measure associated to a current ω is obtained as the probability measure μω corresponding to the measure defined by e-ϕω⁢μref (see [4, Lemma 3.4]).

Definition C.1 ([4, Definition 3.5]).

A finite energy current ω is said to be a (weak) log-Kähler–Einstein metric on the pair (X,Θ) if V-1⁢ωn=μω.

We will use the following functionals on ℰ1:

E⁢(ϕ)=1n+1⁢∑j=0nV-1⁢∫Xϕ⁢ωϕj∧ωrefn-j,

L⁢(ϕ)=-log⁢∫Xe-ϕ⁢μref.

On ℳ1, we will use the functionals

E*⁢(μ)=supϕ∈ℰ1⁡(E⁢(ϕ)-∫Xϕ⁢μ),

H⁢(μ)=∫Xlog⁡(μμref)⁢μ.

The Mabuchi functional 𝐌 and the Ding functional 𝐃 are defined on 𝒯1 by

𝐌⁢(ω)=(H-E*)⁢(V-1⁢ωn),

𝐃⁢(ω)=(L-E)⁢(ϕω).

We say analogously as before that the functional 𝐌 is proper modulo N on smooth K-invariant Kähler metrics if it is bounded from below on smooth K-invariant Kähler metrics and if any sequence (ωi) of smooth K-invariant metrics such that infn∈N⁡J⁢(n⋅ϕωi)→∞ satisfies 𝐌⁢(ωi)→∞.

The definition of the (log-)Mabuchi functional used in Section 7.3.2 differs from the one above in general, but the difference is bounded independently of the metric [7, p. 24, proof of Theorem 4.2], so our definition of coercivity for the log-Mabuchi functional implies properness of 𝐌 modulo Z⁢(G)0 on smooth K-invariant Kähler metrics.

Recall that ℰ1, 𝒯1 and ℳ1 are homeomorphic using the previously defined bijections. For a sequence (ωi) in 𝒯1, strong convergence translates as weak convergence of ωi to some ω∞, together with convergence of J⁢(ϕωi) to J⁢(ϕω∞).

Theorem C.2.

Assume that M is proper modulo N on smooth K-invariant Kähler metrics in c1⁢(X)-Θ. Then there exists a log-Kähler–Einstein metric on the pair (X,Θ).

Proof.

We divide the proof into four steps.

Step 1: Properness provides a candidate log-Kähler–Einstein metric. Indeed, the first item of the properness assumption on 𝐌 implies that it is bounded from below on smooth K-invariant metrics. Let ωi denote a sequence of smooth K-invariant metrics such that 𝐌⁢(ωi) converges to the infimum of 𝐌 on smooth K-invariant metrics. By the second item of the properness assumption and N-invariance, we may as well replace the ωi by ni⁢ωi (for a sequence of ni∈N), so that J⁢(ϕωi) is bounded. It then follows from the comparison between J and E* [4, (1.10)] that E* is bounded on the set {V-1⁢ωin}i. Since 𝐌 and E* are bounded, it follows that the entropy H is bounded on the set {V-1⁢ωin}i. By [4, Theorem 2.17] we can thus replace the sequence (ωi) by a subsequence strongly converging to some ω∞∈𝒯1. Note that since all ωi are K-invariant, the same is true for the limit ω∞.

Step 2: The candidate ω∞ is a minimizer of M in T1K. This claim follows from the fact that the Mabuchi functional on 𝒯1K is the greatest lower semicontinuous extension of its restriction to smooth K-invariant metrics (with respect to the strong topology). Since E* is continuous, it suffices to consider the entropy H. It is lower semicontinuous and the fact that it is the greatest extension of its restriction to smooth K-invariant metrics is proved in [5, Lemma 3.1]. In [5], this result is actually proved without the K-invariance property, but the construction preserves K-invariance, thanks to density of K-invariant smooth functions on X in K-invariant L1-functions on X, and uniqueness in the Calabi–Yau theorem.

Step 3: The Ding functional D is also minimized at ω∞ in T1K. Here we use and imitate [4, Lemma 4.4]. Recall that 𝐌≥𝐃 on 𝒯1 (see [4, Lemma 4.4 i)]). It is thus enough to prove 𝐌⁢(ω∞)≤𝐃⁢(ω) on 𝒯1K. Since ω∞ is a minimizer in 𝒯1K, we have 𝐌⁢(ω∞)≤(H-E*)⁢(μ) for any μ∈ℳ1K. Let ω∈𝒯1K, we have

𝐃⁢(ω)=L⁢(ϕω)-E⁢(ϕω)

=H⁢(μω)+∫Xϕ⁢μω-E⁢(ϕω) ⁢by [4, Section 4.1]

≥𝐌⁢(ω∞)+E*⁢(μω)+∫Xϕ⁢μω-E⁢(ϕω) ⁢since μω is K-invariant

≥𝐌⁢(ω∞),

where the last steps holds by the definition of E*.

Step 4: A minimizer of D in T1K is a log-Kähler–Einstein metric. To prove this claim, we follow [4, proof of Theorem 4.8 ii) ⇒ i)]. The only modification is an argument to pass from K-invariant test functions to arbitrary test functions. Let v∈C0⁢(X)K be a continuous K-invariant function on X. Let ϕt denote the ωref-psh envelope of the function ϕω∞+t⁢v; then it is a function in ℰ1K and [4, proof of Theorem 4.8] shows, by differentiating the Ding functional along t↦ϕt on both sides of 0, that

∫Xv⁢μω∞=V-1⁢∫Xv⁢ω∞n.

Given an arbitrary v∈C0⁢(X), we may apply Fubini’s theorem to the average of v with respect to the invariant probability measure dk on K to obtain

0=∫X(∫Kk⋅v⁢𝑑k)⁢(μω∞-V-1⁢ω∞n)

=∫K(∫X(k⋅v)⁢(μω∞-V-1⁢ω∞n))⁢𝑑k

=∫K(∫Xv⁢(μω∞-V-1⁢ω∞n))⁢𝑑k by K-invariance of ω∞

=∫Xv⁢(μω∞-V-1⁢ω∞n).

As a conclusion, ω∞ is a log-Kähler–Einstein metric on the pair (X,Θ). ∎

Acknowledgements

The several referees for this article deserve special thanks for their valuable comments and corrections. I would also like to warmly thank Chinh Lu who always provided very relevant answers to my questions on several aspects of the variational approach to existence of canonical metrics. The main part of this work was accomplished while I was an FSMP Postdoctoral researcher hosted at the École Normale Supérieure in Paris.

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Received: 2018-03-18

Revised: 2018-12-03

Published Online: 2019-01-20

Published in Print: 2020-06-01

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https://doi.org/10.1515/crelle-2018-0040