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Basis divisors and balanced metrics

  • Yanir A. Rubinstein , Gang Tian and Kewei Zhang EMAIL logo


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.

Award Identifier / Grant number: DMS-1515703

Award Identifier / Grant number: DMS-1906370

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


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


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


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


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


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


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


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:


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


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


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


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


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




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


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


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,


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


So by Theorem A.13, we obtain:

Corollary A.14.

One has


Thus, sending 𝒎(,,),


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




For any effective -divisor D on X, put


Then (see [11])


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


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


More precisely, write


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


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


Definition B.2.

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


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


Theorem B.3.

One has δm(L;θ)=δmA(L;θ).


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,


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,


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,



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


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


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


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).


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


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

When δ=1, we put


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.


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.


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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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