Using log canonical thresholds and basis divisors Fujita–Odaka introduced purely algebro-geometric invariants 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 -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 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 . 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
A.1 Coupled Kähler–Ricci -soliton
We follow the setup of . Let X be a projective manifold. Fix a positive integer k and take a k-tuple of ample -line bundles As in the previous section, assume that there is an effective and holomorphic -action on X. Also assume that this torus action lifts to each . We equip each with a positively curved smooth T-invariant Hermitian metric , whose curvature form will be denoted by . Put , where . Denote by the subspace of consisting of T-invariant Kähler potentials and put
Note that each induces a moment map , whose image will be denoted by . Recall that is a polytope, which does not depend on the choice of . We will fix a smooth positive function for each . Then for any , we have an induced function . 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 Given a k-tuple
we put .
To study the above coupled soliton equations, one needs a coupled Ding function, which we now describe. For each , one can find satisfying
Then it is easy to see that as probability measures, So we can put
Note that μ depends on and α. Now, following , the -weighted θ-Ding functional is defined by
where is the -weighted Monge–Ampère energy (recall (6.2)). Then it is straightforward to check that is θ-twisted coupled Kähler–Ricci -soliton if and only if a critical point of .
As shown in , there are obstructions to the existence of Kähler–Ricci -solitons, mainly coming from the non-coercivity of . So we introduce a coupled coercivity threshold as follows:
Following [7, 47], we can quantize the above setup as follows. Choose a k-tuple of sufficiently divisible integers such that each is very ample. Consider the Bergman space for each pair . We denote by the subspace of T-invariant Bergman potentials in and put
Note that the -action induces a weight decomposition for each vector space . Then proceeding as in Section 6.2 (especially see (6.4)), the -th quantized -weighted Monge–Ampère energy of the pair . Then we put
A.3 Algebraic coupled -weighted -invariant
Motivated by the above formulation, we can now define the algebraic coupled -weighted in the following way.
Following Section 6.3, choose an -basis divisor for each . Summing up, we get a -invariant effective -divisor which will be called an -basis divisor of the k-tuple .
We make the following definitions.
The coupled -weighted -invariant is
When , set
When the torus action is trivial, set
Since any -basis divisor is -invariant, to compute its lct, it suffices to investigate all the -invariant prime divisors F over X. Then (recall (6.8)) one can consider the -th -weighted expected vanishing order of along F.
where F runs through all the -invariant prime divisors over X.
And also, we have the following result.
The number is computed by some -invariant divisor F over X.
The coupled version of Theorem 6.7 also holds.
A.4 Coupled -weighted -balanced metrics
For any , , put
Any critical point of the functional is called coupled -weighted -balanced.
One can think of as the quantization of
being the critical point equations. Varying δ, this can be seen as a continuity path towards the -twisted coupled Kähler–Ricci -soliton metric (cf. ).
We say is coercive on if there exist and such that
If is coercive on , then there exists a coupled -weighted -balanced , minimizing .
The next result is the coupled version of Proposition 4.9.
The functional is coercive on the space if and only if .
A.5 θ-twisted coupled -balanced metric
Recall that we have already chosen a T-invariant smooth form at the beginning. Consider
The next definition gives a natural quantization of the θ-twisted coupled Kähler–Ricci -soliton (see also ).
Any critical point of is called θ-twisted coupled -balanced.
The next result, as a generalization of Theorem 2.3, shows that the coupled -invariant characterizes the existence of θ-twisted coupled -balanced metrics.
The following statements hold:
If , then there exists an θ -twisted coupled -balanced in .
Assume . If there exists an θ -twisted coupled -balanced (respectively a unique θ -twisted coupled -balanced up to translation) in , then (respectively ).
A.6 Computing coupled using -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.
Let X be Kähler manifold, polarized by a k-tuple of ample -line bundles, together with a -action. Put
Let be a k-tuple of positive integers such that each is very ample. Then
Now we apply the above result to the toric Fano case. Assume that X is toric Fano and that Using the toric setup in Section 7, we write , . Up to linear equivalence, we may arrange that for all Each moment polytope is given by
Note that all the weight spaces of are one-dimensional. Then by definition (recall (6.7)) there is only one -basis divisor of each . Thus,
where denotes the -th quantized barycenter of . Hence,
So by Theorem A.13, we obtain:
Thus, sending ,
where denotes the barycenter of . In particular, one always has with equality if and only if Thus, using a result of Hultgren [32, Theorem 2], we obtain the following:
A toric Fano manifold X admits a coupled Kähler–Einstein tuple for if and only if .
B Extension to klt currents
We use the setup of . Let be a triple satisfying:
X is an n-dimensional projective manifold,
θ is a quasi-positive -current, i.e., the sum of a positive current and a smooth form,
θ is klt, i.e., locally writing , then for for some ,
L is an ample -line bundle such that .
For any effective -divisor D on X, put
Then (see )
where denotes the multiplier ideal sheaf associated to the current (and denotes the current of integration along D).
The -invariant of is
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 . Let be a function satisfying
More precisely, write
where is a smooth representative. Let satisfy
Then (up to a constant) . Now define a probability measure on X,
The analytic δ-invariant of is defined by
The analytic -invariant of is defined by
One has .
is proven in much the same way as Corollary 3.3 with two key differences. First, one needs to replace the volume form by the measure , and by . Second, to apply Demailly–Kollár’s theorem  as in the proof of Proposition 3.1, one also needs to invoke the openness [9, 29]. The analogue of Corollary 3.7, namely,
Let be a prime divisor over X. Assume that . Then for any , , and any basis of , there exists such that for any parameter ,
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 , where is a smooth representative. Then it amounts to bounding the integral
from below for any . We pull back everything to Y and work in a polydisc around a very generic point of F, as in the proof of Lemma 3.6. By the definition of , we may further assume that
for some . Recalling (B.1), it suffices to bound
which is a positive quantity only depending on δ, m, , ε and . ∎
Now we turn to balanced metrics. For any , put
A critical point of is called -balanced of level m.
The operator is coercive on if and only if .
After replacing the volume form by the measure and by
the proof goes through following the one for Proposition 4.9. ∎
When , we put
for simplicity and any critical point of is called θ-balanced of level m. Then using Proposition B.6, Berndtsson convexity  and Darvas–Rubinstein principle , we get the following quantized version of [6, Theorem A].
Theorem B.7 (Algebraic characterization of θ-balanced metrics).
One has the following statements:
If there exists a θ -balanced metric of level m.
Suppose that θ is semipositive. If there exists a θ -balanced metric of level m, then . If such a metric is unique, then .
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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