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Basis divisors and balanced metrics

Yanir A. Rubinstein, Gang Tian and Kewei Zhang

Abstract

Using log canonical thresholds and basis divisors Fujita–Odaka introduced purely algebro-geometric invariants δm whose limit in m is now known to characterize uniform K-stability on a Fano variety. As shown by Blum–Jonsson this carries over to a general polarization, and together with work of Berman, Boucksom, and Jonsson, it is now known that the limit of these δm-invariants characterizes uniform Ding stability. A basic question since Fujita–Odaka’s work has been to find an analytic interpretation of these invariants. We show that each δm is the coercivity threshold of a quantized Ding functional on the mth Bergman space and thus characterizes the existence of balanced metrics. This approach has a number of applications. The most basic one is that it provides an alternative way to compute these invariants, which is new even for n. Second, it allows us to introduce algebraically defined invariants that characterize the existence of Kähler–Ricci solitons (and the more general g-solitons of Berman–Witt Nyström), as well as coupled versions thereof. Third, it leads to approximation results involving balanced metrics in the presence of automorphisms that extend some results of Donaldson.

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1515703

Award Identifier / Grant number: DMS-1906370

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11331001

Award Identifier / Grant number: 11890661

Funding statement: The research of Yanir A. Rubinstein was supported by NSF grants DMS-1515703, 1906370 and the Rosi & Max Varon Visiting Professorship at the Weizmann Institute of Science in Fall 2019 and Spring 2020. Gang Tian was supported by NSFC grants 11331001 and 11890661. Kewei Zhang was supported by the CSC award 201706010020 and China post-doctoral grant BX20190014.

A Extension to coupled metrics

In this part we define a coupled δ-invariant for the coupled system of Monge–Ampère equations studied in [33, 32, 20] and record analogues of our main theorems to this more general setting.

A.1 Coupled Kähler–Ricci 𝒈-soliton

We follow the setup of [20]. Let X be a projective manifold. Fix a positive integer k and take a k-tuple of ample -line bundles (L1,,Lk). As in the previous section, assume that there is an effective and holomorphic T-action on X. Also assume that this torus action lifts to each Li. We equip each Li with a positively curved smooth T-invariant Hermitian metric hi, whose curvature form will be denoted by ωi. Put Vi:=Xωin, where 1ik. Denote by ωT(X,ωi) the subspace of (X,ωi) consisting of T-invariant Kähler potentials and put

𝓗T:=ωT(X,ω1)××ωT(X,ωk).

Note that each ωi induces a moment map mωi:Xr, whose image will be denoted by Pi. Recall that Pi is a polytope, which does not depend on the choice of ωic1(Li). We will fix a smooth positive function gi:Pi>0 for each 1ik. Then for any φiωT(X,ωi), we have an induced function gi(φi):=gimωi+ddcφi:X>0. Put

𝒈:=(g1,,gk).

To set up the coupled soliton equations, we need also a twist term, by cohomological reason. Pick a T-invariant smooth form θ in c1(X)-i=1kc1(L). Given a k-tuple

𝝋:=(φ1,,φk)𝓗T,

we put 𝝎𝝋:=(ω1+ddcφ1,,ωi+ddcφk).

Definition A.1 ([33, 32, 20]).

We say that 𝝎𝝋 is an θ-twisted coupled Kähler–Ricci 𝒈-soliton if

Ric(ωi+ddcφi)=j=1k(ωj+ddcφj)+θ+ddcloggi(φi),1ik.

To study the above coupled soliton equations, one needs a coupled Ding function, which we now describe. For each 1ik, one can find fiC(X,) satisfying

Ric(ω)i=i=1kωi+θ+ddcfiandXefiωin=Vi.

Then it is easy to see that as probability measures, ef1ω1nV1==efkωknVk. So we can put

μ:=ef1ω1nV1==efkωknVk.

Note that μ depends on ωi and α. Now, following [20], the 𝒈-weighted θ-Ding functional is defined by

Dα,𝒈(𝝋):=-logXe-iφi𝑑μ-i=1kEωigi(φi),𝝋:=(φ1,,φk)𝓗T,

where Eωigi(φi) is the gi-weighted Monge–Ampère energy (recall (6.2)). Then it is straightforward to check that 𝝎𝝋 is θ-twisted coupled Kähler–Ricci 𝒈-soliton if and only if 𝝋 a critical point of Dα,𝒈.

As shown in [20], there are obstructions to the existence of Kähler–Ricci 𝒈-solitons, mainly coming from the non-coercivity of Dα,𝒈. So we introduce a coupled coercivity threshold as follows:

δA,𝒈(L1,,Lk):=sup{δ>0:sup(φ1,,φk)𝓗TXe-δi=1k(φi-Eωigi(φi))𝑑μ<}.

A.2 Quantization

Following [7, 47], we can quantize the above setup as follows. Choose a k-tuple of sufficiently divisible integers 𝒎:=(m1,,mk) such that each miLi is very ample. Consider the Bergman space mi(X,ωi) for each pair (Li,ωi). We denote by miT(X,ωi) the subspace of T-invariant Bergman potentials in mi(X,ωi) and put

𝓑𝒎T:=m1T(X,ω1)××mkT(X,ωk).

Note that the T-action induces a weight decomposition for each vector space H0(X,miLi). Then proceeding as in Section 6.2 (especially see (6.4)), Eωi,migi() on miT(X,ωi), the mi-th quantized gi-weighted Monge–Ampère energy of the pair (Li,ωi). Then we put

δ𝒎A,𝒈(L1,,Lk):=sup{δ>0:sup(φ1,,φk)𝓑𝒎TXe-δi=1k(φi-Eωi,migi(φi))𝑑μ<}.

A.3 Algebraic coupled 𝒈-weighted δ𝒎-invariant

Motivated by the above formulation, we can now define the algebraic coupled 𝒈-weighted δ𝒎 in the following way.

Following Section 6.3, choose an (mi,gi)-basis divisor DiLi for each Li. Summing up, we get a T-invariant effective -divisor D:=i=1kDii=1kLi, which will be called an (𝒎,𝒈)-basis divisor of the k-tuple (L1,,Lk).

Definition A.2.

We make the following definitions.

  1. The coupled 𝒈-weighted δ𝒎-invariant is

    δ𝒎𝒈(L1,,Lk):=inf{lct(X,D):D is an (𝒎,𝒈)-basis divisor of (L1,,Lk)}.
  2. When 𝒈=(1,,1), set δ𝒎T(L1,,Lk):=δ𝒎𝒈(L1,,Lk).

  3. When the torus action is trivial, set

    δ𝒎(L1,,Lk):=inf{lct(X,i=1kDi):each Di is an mi-basis divisor of Li}.

Since any (𝒎,𝒈)-basis divisor is T-invariant, to compute its lct, it suffices to investigate all the T-invariant prime divisors F over X. Then (recall (6.8)) one can consider Smigi(Li;F), the mi-th gi-weighted expected vanishing order of Li along F.

Lemma A.3.

One has

δ𝒎𝒈(L1,,Lk)=infFAX(F)i=1kSmigi(Li;F),

where F runs through all the TC-invariant prime divisors over X.

And also, we have the following result.

Lemma A.4.

The number δ𝐦𝐠 is computed by some TC-invariant divisor F over X.

The coupled version of Theorem 6.7 also holds.

Theorem A.5.

One has δ𝐦A,𝐠(L1,,Lk)=δ𝐦𝐠(L1,,Lk).

A.4 Coupled 𝒈-weighted (f,δ)-balanced metrics

For any δ>0, fC(X,), put

F𝒎f,δ,𝒈(𝝋):=-1δlogXef-δi=1kφi𝑑μ-i=1kEωi,migi(φi),𝝋=(φ1,,φk)𝓑𝒎T.

Definition A.6.

Any critical point of the functional F𝒎f,δ,𝒈 is called coupled 𝒈-weighted (f,δ)-balanced.

Remark A.7.

One can think of F𝒎f,δ,𝒈 as the quantization of

F𝒎f,δ,𝒈(𝝋):=-1δlogXef-δi=1kφi𝑑μ-i=1kEωigi(φi),𝝋=(φ1,,φk)𝓗T,

with

Ric(ωi+ddcφi)=δj=1k(ωj+ddcφj)+(1-δ)j=1kωj
+(α-ddcf)+ddcloggi(φi),1ik,

being the critical point equations. Varying δ, this can be seen as a continuity path towards the (θ-ddcf)-twisted coupled Kähler–Ricci 𝒈-soliton metric (cf. [20]).

Definition A.8.

We say F𝒎f,δ,𝒈 is coercive on 𝓑𝒎T if there exist λ>0 and C>0 such that

F𝒎f,δ,𝒈(𝝋)λi=1k(supφi-Eωi,migi(φi))-Cfor all 𝝋=(φ1,,φk)𝓑𝒎T.

The next result follows from the argument for Propositions 4.6 and 6.10.

Proposition A.9.

If F𝐦f,δ,𝐠 is coercive on B𝐦T, then there exists a coupled 𝐠-weighted (f,δ)-balanced 𝛗B𝐦T, minimizing F𝐦f,δ,𝐠.

The next result is the coupled version of Proposition 4.9.

Proposition A.10.

The functional F𝐦f,δ,𝐠 is coercive on the space B𝐦T if and only if δ(0,δ𝐦𝐠(L1,,Lk)).

A.5 θ-twisted coupled 𝒈-balanced metric

Recall that we have already chosen a T-invariant smooth form θ(c1(X)-i=1kc1(Li)) at the beginning. Consider

F𝒎θ,𝒈(𝝋):=-logXe-i=1kφi𝑑μ-i=1kEωi,migi(φi),𝝋=(φ1,,φk)𝓑𝒎T.

The next definition gives a natural quantization of the θ-twisted coupled Kähler–Ricci 𝒈-soliton (see also [47]).

Definition A.11.

Any critical point of F𝒎α,𝒈 is called θ-twisted coupled 𝒈-balanced.

The next result, as a generalization of Theorem 2.3, shows that the coupled δ𝒎𝒈-invariant characterizes the existence of θ-twisted coupled 𝒈-balanced metrics.

Theorem A.12.

The following statements hold:

  1. If δ𝒎𝒈(L1,,Lk)>1, then there exists an θ -twisted coupled 𝒈-balanced 𝝋 in 𝓑𝒎T.

  2. Assume θ0. If there exists an θ -twisted coupled 𝒈-balanced 𝝋 (respectively a unique θ -twisted coupled 𝒈-balanced 𝝋 up to translation) in 𝓑𝒎T, then δ𝒎𝒈(L1,,Lk)1 (respectively δ𝒎𝒈(L1,,Lk)>1).

A.6 Computing coupled δm using T-invariant divisors

Note that the proof of Theorem 2.11 also works seamlessly for the coupled case. So we record the following result, without giving the proof.

Theorem A.13.

Let X be Kähler manifold, polarized by a k-tuple (L1,,Lk) of ample Q-line bundles, together with a TC-action. Put

s(L1,,Lk):=sup{s:-KX-si=1kLi is nef}.

Let 𝐦=(m1,,mk) be a k-tuple of positive integers such that each miLi is very ample. Then

min{s(L1,,Lk),δ𝒎T(L1,,Lk)}δ𝒎(L1,,Lk)δ𝒎T(L1,,Lk).

Now we apply the above result to the toric Fano case. Assume that X is toric Fano and that -KX=ikLi. Using the toric setup in Section 7, we write Li=ρaρiDρ, 1ik. Up to linear equivalence, we may arrange that i=1kaρi=1 for all ρ. Each moment polytope Pi is given by

Pi:={uM;u,vρ+aρi0 for all ρ}.

Note that all the weight spaces of H0(X,miLi) are one-dimensional. Then by definition (recall (6.7)) there is only one (mi,1)-basis divisor of each Li. Thus,

δ𝒎T(L1,,Lk)=1maxρ{i=1kbmi(Pi),vρ+1},

where bmi(Pi) denotes the mi-th quantized barycenter of Pi. Hence,

δ𝒎T(L1,,Lk)s(L1,,Lk)=1.

So by Theorem A.13, we obtain:

Corollary A.14.

One has

δ𝒎(L1,,Lk)=δ𝒎T(L1,,Lk)=1maxρ{i=1kbmi(Pi),vρ+1}.

Thus, sending 𝒎(,,),

δ(L1,,Lk):=limδ𝒎(L1,,Lk)=1maxρ{i=1kb(Pi),vρ+1},

where b(Pi) denotes the barycenter of Pi. In particular, one always has δ(L1,,Lk)1, with equality if and only if i=1kb(Pi)=0. Thus, using a result of Hultgren [32, Theorem 2], we obtain the following:

Corollary A.15.

A toric Fano manifold X admits a coupled Kähler–Einstein tuple for (L1,,Lk) if and only if δ(L1,,Lk)=1.

B Extension to klt currents

We use the setup of [6]. Let (X,θ,L) be a triple satisfying:

  1. X is an n-dimensional projective manifold,

  2. θ is a quasi-positive (1,1)-current, i.e., the sum of a positive current and a smooth form,

  3. θ is klt, i.e., locally writing θ=ddcψ, then e-ψLlocp for p[1,1+ε) for some ε>0,

  4. L is an ample -line bundle such that c1(L)=c1(X)-[θ].

Let π:YX be a birational morphism. For any prime divisor FY over X let ordF(θ) be the Lelong number of πθ at a very generic point of F (see [24, 11]). By [6, Lemma 3.3], θ being klt is equivalent to

infFAX(F)ordF(θ)>1.

Set

(B.1)Aθ(F):=AX(F)-ordF(θ).

For any effective -divisor D on X, put

lctθ(X,D):=infFAθ(F)ordF(D).

Then (see [11])

lctθ(X,D)=sup{λ>0:𝒥(θ+λ[D])=𝒪X},

where 𝒥(θ+λ[D]) denotes the multiplier ideal sheaf associated to the current θ+λ[D] (and [D] denotes the current of integration along D).

Definition B.1.

The δm-invariant of (X,θ,L) is

δm(L;θ):=inf{lctθ(X,D):Dm-basis divisor of L}.

Equivalently, one has

δm(L;θ)=infFAθ(F)Sm(F).

In what follows we do not claim nor do we need to know whether the infimum is attained.

Next, we turn to analytic δ-invariants. As before, fix a positively curved smooth Hermitian metric h on L and denote by ω its curvature form, with [ω]=c1(L). Let fθ be a function satisfying

Ric(ω)=ω+θ+ddcfθ,Xefθωn=Xωn=V.

More precisely, write

θ=θ0+ddcψ,

where θ0[θ] is a smooth representative. Let fθ0C(X,) satisfy

Ric(ω)=ω+θ0+ddcfθ0.

Then (up to a constant) fθ=fθ0-ψ. Now define a probability measure on X,

dμθ:=efθωnV.

Definition B.2.

The analytic δ-invariant of (X,θ,L) is defined by

δA(L;θ):=sup{δ>0:supφωXe-δ(φ-E(φ))𝑑μθ<}.

The analytic δm-invariant of (X,θ,L) is defined by

δmA(L;θ):=sup{δ>0:supφmXe-δ(φ-Em(φ))𝑑μθ<}.

Theorem B.3.

One has δm(L;θ)=δmA(L;θ).

Proof.

The proof parallels that of Theorem 2.8 given in Section 3. We point out the main differences for the reader’s convenience.

The proof of Theorem 2.8 amounts to Corollaries 3.3 and 3.7. The analogue of the former, namely,

δm(L;θ)δmA(L;θ),

is proven in much the same way as Corollary 3.3 with two key differences. First, one needs to replace the volume form ωn by the measure dμθ, and AX(F) by Aθ(F). Second, to apply Demailly–Kollár’s theorem [21] as in the proof of Proposition 3.1, one also needs to invoke the openness [9, 29]. The analogue of Corollary 3.7, namely,

δm(L;θ)δmA(L;θ),

requires an extension of Lemma 3.6 to the setting of non-zero Lelong number, given by Lemma B.4 below. ∎

Lemma B.4.

Let FY𝜋X be a prime divisor over X. Assume that ordFθ>0. Then for any δ>0, ε(0,ordFθ), and any basis {si} of H0(X,mL), there exists Cε>0 such that for any parameter t0,

Xet(Aθ(F)+ε)(i=1dmetordF(si)|si|hm2)δm𝑑μθCε>0.

Proof.

We follow the proof of Lemma 3.6, by using a local calculation around a generic point of F, the only difference being that in the presence of the current θ, one should take into account the contribution coming from Lelong numbers.

More precisely, write θ=θ0+ddcψ, where θ0[θ] is a smooth representative. Then it amounts to bounding the integral

Xet(Aθ(F)+ε)e-ψ(i=1dmetordF(si)|si|hm2)δmωn

from below for any t0. We pull back everything to Y and work in a polydisc 𝔻 around a very generic point of F, as in the proof of Lemma 3.6. By the definition of ordFθ, we may further assume that

πψ(ordF(θ)-ε)log|z1|2+Cεon 𝔻

for some Cε>0. Recalling (B.1), it suffices to bound

J(t):=-1|z1|1et(Aθ(F)+ε)|z1|2Aθ(F)+2ε-2(i=1dmetordF(si)|z1|2ordF(si))δm𝑑z1dz1¯
-1|w|1|w|2Aθ(F)+2ε-2(i=1dm|w|2ordF(si))δm𝑑wdw¯,

which is a positive quantity only depending on δ, m, Aθ(F), ε and {ordF(si)}. ∎

Now we turn to balanced metrics. For any δ>0, put

Fmθ,δ(φ):=-1δlogXe-δφ𝑑μθ-Em(φ),φm.

Definition B.5.

A critical point of Fmθ,δ is called (θ,δ)-balanced of level m.

One can also define coercivity for Fmθ,δ as in Definition 4.4. And as in Proposition 4.6, Fmθ,δ being coercive implies the existence of (θ,δ)-balanced metrics of level m.

Proposition B.6.

The operator Fmθ,δ is coercive on Bm if and only if 0<δ<δm(L;θ).

Proof.

After replacing the volume form ωn by the measure dμθ and αm(L) by

αm(L;θ):=sup{λ>0:supφmXe-λ(φ-supφ)𝑑μθ<},

the proof goes through following the one for Proposition 4.9. ∎

When δ=1, we put

Fmθ:=Fmθ,1

for simplicity and any critical point of Fmθ is called θ-balanced of level m. Then using Proposition B.6, Berndtsson convexity [8] and Darvas–Rubinstein principle [19], we get the following quantized version of [6, Theorem A].

Theorem B.7 (Algebraic characterization of θ-balanced metrics).

One has the following statements:

  1. If δm(L;θ)>1 there exists a θ -balanced metric of level m.

  2. Suppose that θ is semipositive. If there exists a θ -balanced metric of level m, then δm(L;θ)1. If such a metric is unique, then δm(L;θ)>1.

Finally, we remark that, with the help of [18, 9, 29], all the results in Section 5 (except Proposition 5.7) and Section 7 can be established in the current setting. One can also extend the above discussions to soliton-type metrics, as in Section 6, and even to coupled soliton metrics, as in Appendix A. But we shall not pursue such generality here.

Acknowledgements

We thank A. Lahdili and F. Wang for helpful discussions. K. Zhang was a Visiting Scholar at the University of Maryland in 2017–2018 when part of this work was initiated and is grateful to T. Darvas for many inspiring conversations back then.

References

[1] F. Ambro, Variation of log canonical thresholds in linear systems, Int. Math. Res. Not. IMRN 2016 (2016), no. 14, 4418–4448. 10.1093/imrn/rnv284Search in Google Scholar

[2] T. Aubin, Réduction du cas positif de l’équation de Monge–Ampère sur les variétés kählériennes compactes à la démonstration d’une inégalité, J. Funct. Anal. 57 (1984), no. 2, 143–153. 10.1016/0022-1236(84)90093-4Search in Google Scholar

[3] R. J. Berman, Relative Kähler–Ricci flows and their quantization, Anal. PDE 6 (2013), no. 1, 131–180. 10.2140/apde.2013.6.131Search in Google Scholar

[4] R. J. Berman, S. Boucksom, P. Eyssidieux, V. Guedj and A. Zeriahi, Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties, J. reine angew. Math. 751 (2019), 27–89. 10.1515/crelle-2016-0033Search in Google Scholar

[5] R. J. Berman, S. Boucksom, V. Guedj and A. Zeriahi, A variational approach to complex Monge-Ampère equations, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 179–245. 10.1007/s10240-012-0046-6Search in Google Scholar

[6] R. J. Berman, S. Boucksom and M. Jonsson, A variational approach to the Yau–Tian–Donaldson conjecture, preprint (2018), https://arxiv.org/abs/1509.04561v2. 10.1090/jams/964Search in Google Scholar

[7] R. J. Berman and D. Witt Nystrom, Complex optimal transport and the pluripotential theory of Kähler–Ricci solitons, preprint (2014), https://arxiv.org/abs/1401.8264. Search in Google Scholar

[8] B. Berndtsson, A Brunn–Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry, Invent. Math. 200 (2015), no. 1, 149–200. 10.1007/s00222-014-0532-1Search in Google Scholar

[9] B. Berndtsson, The openness conjecture and complex Brunn–Minkowski inequalities, Complex geometry and dynamics, Abel Symp. 10, Springer, Cham (2015), 29–44. 10.1007/978-3-319-20337-9_2Search in Google Scholar

[10] H. Blum and M. Jonsson, Thresholds, valuations, and K-stability, Adv. Math. 365 (2020), 107062. 10.1016/j.aim.2020.107062Search in Google Scholar

[11] S. Boucksom, C. Favre and M. Jonsson, Valuations and plurisubharmonic singularities, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 449–494. 10.2977/prims/1210167334Search in Google Scholar

[12] D. Catlin, The Bergman kernel and a theorem of Tian, Analysis and geometry in several complex variables (Katata 1997), Trends Math., Birkhäuser, Boston (1999), 1–23. 10.1007/978-1-4612-2166-1_1Search in Google Scholar

[13] I. A. Cheltsov, Y. A. Rubinstein and K. Zhang, Basis log canonical thresholds, local intersection estimates, and asymptotically log del Pezzo surfaces, Selecta Math. (N. S.) 25 (2019), no. 2, Paper No. 34. 10.1007/s00029-019-0473-zSearch in Google Scholar

[14] I. A. Cheltsov and K. A. Shramov, Log-canonical thresholds for nonsingular Fano threefolds, Uspekhi Mat. Nauk 63 (2008), no. 5(383), 73–180. Search in Google Scholar

[15] X. Dai, K. Liu and X. Ma, On the asymptotic expansion of Bergman kernel, J. Differential Geom. 72 (2006), no. 1, 1–41. 10.4310/jdg/1143593124Search in Google Scholar

[16] T. Darvas, The Mabuchi geometry of finite energy classes, Adv. Math. 285 (2015), 182–219. 10.1016/j.aim.2015.08.005Search in Google Scholar

[17] T. Darvas, Geometric pluripotential theory on Kähler manifolds, Advances in complex geometry, Contemp. Math. 735, American Mathematical Society, Providence (2019), 1–104. 10.1090/conm/735/14822Search in Google Scholar

[18] T. Darvas, C. H. Lu and Y. A. Rubinstein, Quantization in geometric pluripotential theory, Comm. Pure Appl. Math. 73 (2020), no. 5, 1100–1138. 10.1002/cpa.21857Search in Google Scholar

[19] T. Darvas and Y. A. Rubinstein, Tian’s properness conjectures and Finsler geometry of the space of Kähler metrics, J. Amer. Math. Soc. 30 (2017), no. 2, 347–387. 10.1090/jams/873Search in Google Scholar

[20] T. Delcroix and J. Hultgren, Coupled complex Monge–Ampère equations on Fano horosymmetric manifolds, preprint (2018), https://arxiv.org/abs/1812.07218. 10.1016/j.matpur.2020.12.002Search in Google Scholar

[21] J.-P. Demailly and J. Kollár, Semi-continuity of complex singularity exponents and Kähler–Einstein metrics on Fano orbifolds, Ann. Sci. Éc. Norm. Supér. (4) 34 (2001), no. 4, 525–556. 10.1016/S0012-9593(01)01069-2Search in Google Scholar

[22] S. K. Donaldson, Scalar curvature and projective embeddings. I, J. Differential Geom. 59 (2001), no. 3, 479–522. 10.4310/jdg/1090349449Search in Google Scholar

[23] S. K. Donaldson, Scalar curvature and projective embeddings. II, Q. J. Math. 56 (2005), no. 3, 345–356. 10.1093/qmath/hah044Search in Google Scholar

[24] C. Favre and M. Jonsson, Valuative analysis of planar plurisubharmonic functions, Invent. Math. 162 (2005), no. 2, 271–311. 10.1007/s00222-005-0443-2Search in Google Scholar

[25] K. Fujita, Uniform K-stability and plt blowups of log Fano pairs, Kyoto J. Math. 59 (2019), no. 2, 399–418. 10.1215/21562261-2019-0012Search in Google Scholar

[26] K. Fujita and Y. Odaka, On the K-stability of Fano varieties and anticanonical divisors, Tohoku Math. J. (2) 70 (2018), no. 4, 511–521. 10.2748/tmj/1546570823Search in Google Scholar

[27] W. Fulton, Introduction to toric varieties, Ann. of Math. Stud. 131, Princeton University, Princeton 1993. 10.1515/9781400882526Search in Google Scholar

[28] A. Golota, Delta-invariants for Fano varieties with large automorphism groups, Internat. J. Math. 31 (2020), no. 10, Article ID 2050077. 10.1142/S0129167X20500779Search in Google Scholar

[29] Q. Guan and X. Zhou, A proof of Demailly’s strong openness conjecture, Ann. of Math. (2) 182 (2015), no. 2, 605–616. 10.4007/annals.2015.182.2.5Search in Google Scholar

[30] J. Han and C. Li, On the Yau–Tian–Donaldson conjecture for generalized Kähler–Ricci soliton equations, preprint (2020), https://arxiv.org/abs/2006.00903. Search in Google Scholar

[31] Y. Hashimoto, Mapping properties of the Hilbert and Fubini-Study maps in Kähler geometry, Ann. Fac. Sci. Toulouse Math. (6) 29 (2020), no. 2, 371–389. 10.5802/afst.1635Search in Google Scholar

[32] J. Hultgren, Coupled Kähler–Ricci solitons on toric Fano manifolds, Anal. PDE 12 (2019), no. 8, 2067–2094. 10.2140/apde.2019.12.2067Search in Google Scholar

[33] J. Hultgren and D. W. Nyström, Coupled Kähler–Einstein metrics, Int. Math. Res. Not. IMRN 2019 (2019), no. 21, 6765–6796. 10.1093/imrn/rnx298Search in Google Scholar

[34] L. Ioos, Anticanonically balanced metrics on fano manifolds, preprint (2020), https://arxiv.org/abs/2006.05989. 10.1007/s10455-022-09834-4Search in Google Scholar

[35] S. Kobayashi, Differential geometry of complex vector bundles, Publ. Math. Soc. Japan 15, Princeton University, Princeton 1987. 10.1515/9781400858682Search in Google Scholar

[36] J. Kollár, Singularities of pairs, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math. 62, American Mathematical Society, Providence (1997), 221–287. 10.1090/pspum/062.1/1492525Search in Google Scholar

[37] A. Lahdili, Kähler metrics with constant weighted scalar curvature and weighted K-stability, Proc. Lond. Math. Soc. (3) 119 (2019), no. 4, 1065–1114. 10.1112/plms.12255Search in Google Scholar

[38] C. Li, Greatest lower bounds on Ricci curvature for toric Fano manifolds, Adv. Math. 226 (2011), no. 6, 4921–4932. 10.1016/j.aim.2010.12.023Search in Google Scholar

[39] C. Li and C. Xu, Stability of valuations: Higher rational rank, Peking Math. J. 1 (2018), no. 1, 1–79. 10.1007/s42543-018-0001-7Search in Google Scholar

[40] T. Mabuchi, Some symplectic geometry on compact Kähler manifolds. I, Osaka J. Math. 24 (1987), no. 2, 227–252. Search in Google Scholar

[41] J. Park and J. Won, K-stability of smooth del Pezzo surfaces, Math. Ann. 372 (2018), no. 3–4, 1239–1276. 10.1007/s00208-017-1602-7Search in Google Scholar

[42] W.-D. Ruan, Canonical coordinates and Bergmann metrics, Comm. Anal. Geom. 6 (1998), no. 3, 589–631. 10.4310/CAG.1998.v6.n3.a5Search in Google Scholar

[43] Y. A. Rubinstein, Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics, Adv. Math. 218 (2008), no. 5, 1526–1565. 10.1016/j.aim.2008.03.017Search in Google Scholar

[44] Y. A. Rubinstein, On the construction of Nadel multiplier ideal sheaves and the limiting behavior of the Ricci flow, Trans. Amer. Math. Soc. 361 (2009), no. 11, 5839–5850. 10.1090/S0002-9947-09-04675-3Search in Google Scholar

[45] Y. Shi, On the α-invariants of cubic surfaces with Eckardt points, Adv. Math. 225 (2010), no. 3, 1285–1307. 10.1016/j.aim.2010.03.024Search in Google Scholar

[46] G. Székelyhidi and V. Tosatti, Regularity of weak solutions of a complex Monge–Ampère equation, Anal. PDE 4 (2011), no. 3, 369–378. 10.2140/apde.2011.4.369Search in Google Scholar

[47] R. Takahashi, Geometric quantization of coupled Kähler–Einstein metrics, preprint (2019), https://arxiv.org/abs/1904.12812. 10.2140/apde.2021.14.1817Search in Google Scholar

[48] G. Tian, On Kähler–Einstein metrics on certain Kähler manifolds with C1(M)>0, Invent. Math. 89 (1987), no. 2, 225–246. 10.1007/BF01389077Search in Google Scholar

[49] G. Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom. 32 (1990), no. 1, 99–130. 10.4310/jdg/1214445039Search in Google Scholar

[50] G. Tian, On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990), no. 1, 101–172. 10.1007/BF01231499Search in Google Scholar

[51] G. Tian, On stability of the tangent bundles of Fano varieties, Internat. J. Math. 3 (1992), no. 3, 401–413. 10.1142/S0129167X92000175Search in Google Scholar

[52] G. Tian and X. Zhu, A new holomorphic invariant and uniqueness of Kähler–Ricci solitons, Comment. Math. Helv. 77 (2002), no. 2, 297–325.10.1007/s00014-002-8341-3Search in Google Scholar

[53] C. Xu and Z. Zhuang, Uniqueness of the minimizer of the normalized volume function, preprint (2020), https://arxiv.org/abs/2005.08303. 10.4310/CJM.2021.v9.n1.a2Search in Google Scholar

[54] S. Zelditch, Szegő kernels and a theorem of Tian, Int. Math. Res. Not. IMRN 1998 (1998), no. 6, 317–331. 10.1155/S107379289800021XSearch in Google Scholar

[55] K. Zhang, Continuity of delta invariants and twisted Kähler–Einstein metrics, preprint (2020), https://arxiv.org/abs/2003.11858. 10.1016/j.aim.2021.107888Search in Google Scholar

[56] X. Zhu, Kähler–Ricci soliton typed equations on compact complex manifolds with C1(M)>0, J. Geom. Anal. 10 (2000), no. 4, 759–774. 10.1007/BF02921996Search in Google Scholar

[57] Z. Zhuang, Optimal destabilizing centers and equivariant K-stability, preprint (2020), https://arxiv.org/abs/2004.09413. 10.1007/s00222-021-01046-0Search in Google Scholar

Received: 2020-11-09
Revised: 2021-02-01
Published Online: 2021-04-29
Published in Print: 2021-09-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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