A non-Archimedean approach to K-stability, II: Divisorial stability and openness
-
Sébastien Boucksom
and
Mattias Jonsson
Abstract
To any projective pair
Introduction
Consider, for the moment, a complex projective manifold X, equipped with an ample
As originally pointed out in [52], (ample) test configurations for
and the missing part of the (uniform) YTD conjecture thus consists in proving the (purely algebro-geometric) statement that uniform K-stability for filtrations already follows from its version for test configurations, viewed as filtrations of finite type.
Our primary goal in this paper is to provide a valuative characterization of uniform K-stability for filtrations, as used by Li in [39]. Using this, we will prove:
Main Theorem.
Uniform K-stability for filtrations is an open condition on the polarization.
More precisely, the condition of
To explain the proof of the main theorem, let us first consider the Fano case
Crucially, the MMP techniques of [41] further show that K-stability of a Fano manifold (or, more generally, any log Fano pair) can be tested using only “special” test configurations, which correspond to so-called “dreamy” divisorial valuations [30]; this leads to a purely valuative characterization of K-stability [38, 30], which has been instrumental in the recent spectacular progress towards the deeper understanding of K-stability of log Fano varieties and the construction of moduli spaces thereof; see [7, 43], to name just a few.
Motivated by this, R. Dervan and E. Legendre initiated in [27] the study of a valuative criterion in the case of a general polarization, by providing a purely valuative expression
In contrast with the log Fano case, valuative stability is not expected to imply K-stability for a general polarization, and the MMP will likely play a less important role in that case as well. Using our previous works [20, 19], we will extend the β-invariant to convex combinations of divisorial valuations, viewed as atomic measures on the Berkovich analytification of X, and show that the corresponding stability notion, which we call divisorial stability is equivalent to (uniform) K-stability for filtrations.
Valuative stability
As in the main body of the paper, we consider from now an arbitrary polarized pair
Before describing our notion of divisorial stability, let us first briefly revisit the definitions of [27, 42] in the present setting.
Denote by
This quantity, which appeared in [44, 38, 30] under various normalizations and notation, coincides with the expected vanishing order
Using the differentiability of the volume function on the big cone [12, 36], one checks that
where
The β-invariant defines a
Divisorial stability
Our stability notion involves the larger space
where
There is an obvious embedding
this does not depend on the class ω. To extend the second term in (0.3), we interpret the invariant (0.2) as the energy of the Dirac mass
To explain this, recall first that the space
where
Each
for
As mentioned before, the energy
When
Our first main result is as follows:
Theorem A.
For any measure
The result actually holds for all measures of finite energy, i.e.
Given
for the directional derivative of the energy with respect to ω. By approximation, this is again fully characterized by the case where
For any polarized pair
We say that
When
Divisorial stability threshold and openness
As in [42], it is convenient to introduce the divisorial stability threshold of the polarized pair
Thus
Theorem B.
For any polarized pair
Recall that the pair
Corollary C.
If a polarized pair
The last part of Corollary C, together with Theorem D below, implies the Main Theorem above.
The first part can be viewed as a version in our context of a celebrated result of Y. Odaka [45] (see also [15]), to the effect that any polarized pair
Part (ii) of Theorem B is a consequence of a refined version of Theorem A, involving Hölder estimates for the directional derivative
In the log Fano case, the divisorial stability threshold essentially reduces to the δ-invariant defined in [32, 5]. In fact, for any sublc polarized pair
By the linearity of the entropy and convexity of the energy, this implies
where
coincides with the usual δ-invariant. As a consequence, divisorial stability, (uniform) valuative stability, and uniform K-stability are all equivalent in the log Fano case (see Corollary 4.11).
K-stability for filtrations
Next we discuss the relation between divisorial stability and the stability notions by C. Li [39]. We note that Li assumed that X is smooth, as he relied on the preprint [17], and specifically continuity of envelopes, which is not yet known in the singular case. Using [19], we are able to bypass this obstacle.
Consider thus a polarized pair
where the last equality holds by homogeneity with respect to the scaling action of
Assume further
where
The Rees construction yields an identification of the subset
Theorem D.
The divisorial stability threshold of any polarized subklt pair
In particular,
In view of (0.5), the main step in the proof of Theorem D consists in showing that the image of
where
As a consequence, the infimum in Theorem D can be computed on the space
A fortiori, divisorial stability implies uniform K-stability, as we already noted above. The two notions are equivalent if a certain entropy regularization conjecture holds, as first formulated in [17]. A stronger, and more precise conjecture, goes as follows:
Conjecture.
Let
Granted this conjecture, divisorial stability is the same as uniform K-stability – and the uniform YTD conjecture thus holds for any polarized complex manifold, as discussed in the beginning of the introduction. See also [40].
In the Fano case, divisorial stability and valuative stability are equivalent. In the general case, we do not expect this to be true, but one can ask whether it is enough to consider divisorial measures with a fixed bound on the cardinality of their supports. For example, [28, Proposition 5.3.1] shows that on a toric surface, it suffices to consider measures supported on a set of cardinality at most two.
Structure of the paper
The article is organized as follows.
Finally, Appendix A reviews the properties of the log discrepancy function of a pair, and its extension to the Berkovich space in the subklt case.
1 Preliminaries
The main purpose of this preliminary section is to recall results from non-Archimedean pluripotential theory, as developed in [20], which form the building blocks of our approach.
1.1 Notation and conventions
A function
A net in a set Z is a family
If Z is a Hausdorff topological space, and
For
1.2 Quasi-metric spaces
A quasi-metric on a set Z is a function
for some constant
A continuous function
Lemma 1.1.
Let
Proof.
Uniqueness is clear, by density of D. Pick
1.3 Positive numerical classes
In the entire paper, we work over an algebraically closed field k of characteristic 0, and X denotes an irreducible projective variety over k, i.e. an integral projective k-scheme (not necessarily normal for now). We set
Denote by
Each
We will occasionally use the Thompson metric of the open convex cone
It is locally equivalent to the metric on
The volume of
We define the trace of
Note that
The trace computes the logarithmic derivative of the volume, i.e.
Example 1.2.
When
The volume function
is continuous, positive precisely on the open convex cone
for all
1.4 Berkovich analytification and psh functions
The Berkovich analytification
The space
The subspace
Denote by
The space
To each
defined in [20, Section 4], and characterized as follows:
a PL function φ, written
each
Each
A simple approximation argument yields
The (Borel) set
is independent of ω. It is contained in
Example 1.3.
If
and hence
Example 1.4.
If
In particular, every ω-psh function is finite-valued on
1.5 Energy pairing and functions of finite energy
The energy pairing is first defined as a symmetric
to
where
Further, if
For any
This normalization guarantees the equivariance property
for
The space of ω-psh functions of finite energy is defined as
and the strong topology of
The vector space
Theorem 1.5.
The energy pairing
The energy pairing is further homogeneous with respect to the scaling action (1.12), i.e.
for
Since
a mixed Monge–Ampère measure
by requiring that
for all
Fix now
By (1.19), we further have
for
We also get a Monge–Ampère operator
If
for all
for any
1.6 Measures of finite energy
Fix
A maximizing net for μ is defined as a net
for any
The energy functional
The space
of the energy functional. By definition, any
In contrast with
for all
Example 1.6.
For any
When
As in the case of
By Example 1.6, the Monge–Ampère operator defines a map
with respect to the strong topology on both sides. Further, a net
The Monge–Ampère operator maps
We define the normalized potential of a measure μ in the image of
By (1.29), we have
and
For any
Example 1.7.
For any
Then
where
see (1.11).
In [20] and [19], it was respectively shown that the strong topology of
Theorem 1.8.
There exists a unique strongly continuous functional
such that
for all
Finally, for all
with
Proof of Theorem 1.8.
By [20, Section 10], Theorem 1.8 is valid for a certain quasi-metric
Next, note that uniqueness is clear, by density of the image of
for all
Using
with
2 Differentiability of the energy of a measure
The main purpose of this section is to establish the differentiability of the energy of a measure with respect to the ample class (Theorem 2.15 below, which corresponds to Theorem A in the introduction), along with Hölder estimates for the derivative (Theorem 2.12) that will be the main ingredient for the continuity of the divisorial stability threshold (Theorem B (ii) in the introduction).
As before, X denotes an arbitrary (possibly non-normal) projective variety, of dimension n, defined over an algebraically closed field k (whose characteristic can in fact be arbitrary in this section).
2.1 The twisted Monge–Ampère energy
Fix for the moment
Definition 2.1.
We define the θ-twisted Monge–Ampère energy
The right-hand side involves the energy pairing, see Theorem 1.5. By (1.15), we have
where
(see Section 1.3). Further,
for
As with the Monge–Ampère energy
Proposition 2.2.
For all
and
for all
As a special case of [20, Theorem 3.25], we also have:
Lemma 2.3.
If θ is nef, then the restriction of
The twisted energy is closely related to the differential of
Lemma 2.4.
For each
given by
Note that (2.6) is translation invariant as a function of φ (see (2.1) and (1.18)).
Proof.
Note that smoothness follows from the fact that
Combined with (1.7), this yields
which proves (2.6). ∎
By (2.2) (or by differentiating (1.22) with respect to ω), we have
for
Lemma 2.5.
For any
2.2 The twisted energy of a measure
Fix a class
Proposition--Definition 2.6.
For any
for all
for all
By uniqueness,
Remark 2.7.
When the class ω has the envelope property (e.g. when X is smooth), the Monge–Ampère operator induces a homeomorphism
Lemma 2.8.
The exists
for all
Proof.
By homogeneity, we may first assume
and
with
Proof of Proposition 2.6.
Pick measures
see Theorem 1.8. This proves (2.9) when
Proposition 2.9.
For all
Proof.
By continuity, we may assume that
with
see (2.7). ∎
Note also that
for each
Remark 2.10.
If we take for granted Theorem 2.15, then (2.13) can alternatively be obtained by differentiating (1.27), while (2.14) becomes the Euler equation reflecting the homogeneity property
For later use, we also note:
Lemma 2.11.
If μ lies in the image of
2.3 Hölder continuity of the energy of a measure
The next result will be the main ingredient leading to the continuity of the divisorial stability threshold (see Theorem 4.12).
Theorem 2.12.
Suppose that
for all
Combined with (1.28), this implies that
Lemma 2.13.
Suppose
The proof relies on the following variant of [20, Lemma 9.25].
Lemma 2.14.
Assume given
Proof.
The first estimates are already proved in [20, Lemma 9.25], and we simply repeat the argument to get the remaining ones. Since
By the concavity of
and hence
The result follows after dividing by
Proof of Lemma 2.13.
Note that the desired estimates are invariant under translation of φ. To prove (2.16), (2.17) and (2.18), we first normalize φ by
Further,
Similarly, Lemma 2.14 yields
By (2.11) and (1.6), we further have
By (2.6), this yields
and (2.17) follows.
Finally, (2.6) implies
and (2.18) thus follows from (2.16) and (2.17) together with
Next we turn to (2.19) and (2.20), for which we now normalize φ by
see Theorem 1.8 (iv). By (2.16) we further have
which yields (2.19) thanks to (2.21) and (1.28). Finally,
see Theorem 1.8 (iv). By (1.28), (2.16) and (2.21), this yields
where the implicit constant in O only depends on n. This proves (2.20). ∎
Proof of Theorem 2.12.
Note first that
Arguing successively with
so that
On the other hand, (2.9), (2.19) and (2.20) yield
with
2.4 Differentiability of the energy with respect to the class
Theorem 2.15.
For any
for all
Lemma 2.16.
Pick a class
Then
for a constant
Proof.
We may assume that
By the density of the image of
On the other hand, (2.22) implies
and hence
with
On the one hand, we have
On the other hand, injecting (2.23) into [20, Corollary 7.35] yields, for
where
where the implicit constant in O only depends on A and n. Since
see (1.7), we infer
using (2.6), (2.8), (2.23) as well as
Proof of Theorem 2.15.
Pick
and
where
as
This proves that
Remark 2.17.
Assume that all ample classes on X have the envelope property (e.g. X is smooth). Then any
Since
2.5 The case of a Dirac mass
In this subsection, we fix
for
In what follows, we provide explicit formulas for these invariants when
Theorem 2.18.
Pick a prime divisor
For simplicity, we have set
see Section 1.3 for the notation. Recall from (1.11) that
Up to the factor
Proof of Theorem 2.18.
By (1.28), the left-hand side of (2.27) is continuous with respect to ω. The same holds for the right-hand side, using
and the continuity
with
for
and a simple computation thus shows that (2.28) is equivalent to
While this appears to be a simple differentiation under the integral sign, the slight twist here is that
which proves that
Next, we may assume that
and it remains to see
as
3 Entropy and the δ-invariant
In what follows,
3.1 Divisorial measures
Definition 3.1.
We define a divisorial measure on
A divisorial measure
for a finite set
As
Example 3.2.
For any
Lemma 3.3.
The space
Proof.
For any
3.2 The entropy functional of a pair
Following [15], we define the entropy functional
of the pair
for all
Remark 3.4.
When
The entropy functional is clearly affine on the convex set
if
for
3.3 The δ-invariant
Consider as above an arbitrary polarized pair
Definition 3.5.
The δ-invariant of
Note that
for
Theorem 3.6.
For any polarized pair
if
(3.4)if
where
Note that statement (iv) implies in particular that
When
In what follows, we write for simplicity
Lemma 3.7.
For any
for a constant
Proof.
Pick a maximizing sequence
strongly in
with
and
with
Letting
By the convexity of the energy functional
We thus get
Proof of Theorem 3.6.
Assume
for
with
Assume now
Next, assume that
by the convexity of
3.4 Extending the entropy functional
Assume in this subsection that the pair
see Appendix A. The entropy functional thus admits a natural extension
defined by
for any
for any
Note also that
Example 3.9.
Assume
By [20, Example 3.19], we have
As
The following approximation result, which can be viewed as a non-Archimedean analogue of [3, Lemma 3.1], plays a key role in what follows.
Theorem 3.10.
Any
When X is smooth and
Corollary 3.11.
The δ-invariant of any polarized subklt pair
Proof.
By the definition of
We will rely several times on the following simple observation:
Lemma 3.12.
Let
Proof.
Since f is lsc, we have
The next two results are the key ingredients in the proof of Theorem 3.10.
Lemma 3.13.
Assume X is smooth. Let
any
for any weakly convergent net
Here
Proof.
Assertions (i) and (ii) follow from [20, Theorems A.1 and A.4] (the latter being a consequence of the uniform Lipschitz estimates of [14, Theorem C]). Now consider a convergent net
Lemma 3.14.
Assume X is smooth. Pick
for each snc test configuration
Proof.
Any
for all
Since
Proof of Theorem 3.10.
Observe first that the result is trivial if
We therefore assume
To see this, pick a resolution of singularities
For any snc test configuration
By Lemma 3.14, we have
By Lemma 3.12, we infer
Replacing μ with
On the one hand, by the continuity of
uniformly with respect to
3.5 The δ-invariant and Ding-stability
As in the previous subsection, we assume that
For any function
The (non-Archimedean) Ding functional
of the polarized pair
where
for
Proposition 3.15.
The Ding functional is finite-valued, and satisfies a Hölder estimate
for all
Proof.
By the translation invariance of the Ding functional (see (3.14)), we may assume that
for each
Thus
and the same holds for ψ. Now set
with
As we next show, the δ-invariant is a threshold for Ding-stability (see also [2, 6]).
Theorem 3.16.
Set
By the strong continuity of the Ding functional (see Proposition 3.15), it is enough to test the conditions in the second halves of (i) and (ii) on the dense subset
Proof.
Assume first that
where the first term tends to
and hence
see [20, Lemma 7.33] for the last inequality. As
4 Divisorial stability
As before,
4.1 The β-functional
We now introduce the key functional in this paper.
Definition 4.1.
The β-functional
for
To simplify notation, we slightly abusively use
Note that
for all
The β-functional is closed related to the (non-Archimedean) Mabuchi K-energy functional of
by setting
for
The main advantage of β over
Consider next
compare (2.25). Then
Lemma 4.2.
For any given prime divisor F on a smooth birational model
Since
Assume, finally, that the pair
defined by (4.2) for any
The Mabuchi K-energy also admits an extension
defined by (4.4) for
4.2 Divisorial stability
We are now in a position to introduce the main new concept in this paper.
Definition 4.3.
For any polarized pair
we say that
divisorially semistable if
divisorially stable if
As noted above, Lemma 4.2 shows that the restriction of β to
Lemma 4.5.
If the polarized pair
As we next show, divisorial stability also implies the usual notion of K-stability (see Section 5.5 for a more detailed discussion).
Lemma 4.6.
Assume
Proof.
By [20, Corollary 2.32], the set of normal, ample test configurations
When
According to an important result of Odaka [45] (see also [15, Theorem 9.1]), any pair
Corollary 4.7.
If the polarized pair
4.3 Divisorial stability threshold and openness
Definition 4.8.
We define the divisorial stability threshold of a polarized pair
By definition, we thus have
Note also that
for
As in Corollary 4.7, Theorem 3.6 (i) and (2.10) directly imply:
Proposition 4.9.
For any polarized pair
and we then further have
In particular, if
This proves Theorem B (i) and the first part of Corollary C in the introduction.
In line with [46] (see also [15, Section 9.1]), we next show:
Proposition 4.10.
Let
Further:
if
if
if
Proof.
By (2.14) and the linearity of
and the β-functional thus takes the simpler form
This directly implies (4.10), and the rest follows from Theorem 3.6. ∎
Corollary 4.11.
For any polarized subklt pair
If we further assume that B is effective, i.e.
Proof.
Assertions (i) and (ii) are direct consequences of Theorem 3.16 and Proposition 4.10, while the final assertion follows from [30, Corollary 6.11]. ∎
We may now state one of the main results of this article.
Theorem 4.12.
If
Proof.
Set, for simplicity,
By (4.9) and Theorem 3.6 (iv),
of the open convex cone
for a constant
we get
On the other hand, (1.28) yields
and hence
Since
Remark 4.13.
As in [42], one can introduce the valuative stability threshold
Restricting the above argument to
Remark 4.14.
Assume
where the right-hand equality holds by homogeneity of β and
5 Comparison to other stability notions
In this final section, we consider a polarized pair
5.1 Norms and filtrations
We use [19] as a reference for what follows. For any
We denote by
The space
The space
The Monge–Ampère operator
is the unique continuous map (with respect to the
Extending [26], we define the minimum norm of
of the Monge–Ampère measure
Any
for a finite subset
The subset
where the left-hand side is equipped with the quotient topology and the right-hand side with the strong topology. In particular, a divisorial norm
Recall from [19, Section 2.2] that a norm
To any (not necessarily ample, nor normal) test configuration
Lemma 5.1.
We have
Proof.
For any test configuration
5.2 Divisorial stability in terms of divisorial norms
Consider the functional
of the polarized pair
for
in line with (4.5), we define the Mabuchi K-energy functional
by setting
Then
for all
Theorem 5.2.
The divisorial stability threshold of any polarized pair
In particular, the pair
Recall that a divisorial norm
Lemma 5.3.
For any divisorial norm
Proof.
Pick a finite subset
see [19, (6.2)]. By the continuity of
strongly in
Thus
Thus
and we are done. ∎
Proof of Theorem 5.2.
For any
Since
Using Lemma 5.3, it is further easy to see that the latter infimum can be computed using norms in
Remark 5.4.
In the work [29], K. Fujita defined a notion of divisorial (semi)stability for a
5.3 K-stability for filtrations
In this subsection, we assume that
Definition 5.5.
We define the Mabuchi K-energy functional
of the polarized subklt pair by setting
for any
When
By the continuity of
Lemma 5.6.
For any norm
Proof.
By Lemma 5.3, it is enough to produce such a sequence in
As
Definition 5.7.
A polarized subklt pair
Theorem 5.8.
The divisorial stability threshold of any polarized subklt pair
In particular,
Note that Theorem 5.8 and Theorem 4.12 together imply the Main Theorem stated in the beginning of the introduction.
Proof.
Set
see Remark 4.14. Thus
while the other inequality follows from Theorem 5.2. Finally, the last equality holds by homogeneity with respect to the scaling action of
Remark 5.9.
The analogue of the last expression involving the “unit sphere”
In view of Corollary 4.11, we also get the following result in the log Fano case.
Corollary 5.10.
For any polarized subklt pair
5.4 K-stability for models
Let
Proposition 5.11.
The divisorial stability threshold satisfies
In particular,
This means that divisorial stability is equivalent to Li’s notion of uniform K-stability for models introduced in [39].[10] Indeed, a (normal) test configuration is called a model in [39]. Li assumes X is smooth as his definition relies on continuity of envelopes as in [17], but we can bypass this using [19]. We note that [39, Proposition 6.3] can be used to show that uniform K-stability for models is equivalent to uniform K-stability for filtrations, in the setting considered there.
5.5 Divisorial stability vs. K-stability
We return to the setting of an arbitrary polarized pair
Let
where
Homogenization
Recall that the Rees construction yields an identification of the set
To each
(see for instance [15, Definition 3.17]), and
K-semistable if
uniformly K-stable if there exists
In analogy with Definition 4.8 (see also [8, Theorem 1.1]), we introduce:
Definition 5.12.
The K-stability threshold of
Thus
Theorem 5.13.
The K-stability threshold of any polarized pair
In particular,
Lemma 5.14.
Any
Proof.
Let
Thus we have
Proof of Theorem 5.13.
Set
For any
Since
We conjecture that equality holds in Theorem 5.13:
Conjecture 5.15.
For any polarized pair
In particular,
Note that the last part holds in the log Fano case, see Corollary 4.11.
Assuming
where uniquely cscK means that
Conjecture 5.15 would in turn follow from the following:
Conjecture 5.16.
Pick
Conjecture 5.16 can be viewed as a more precise version of [17, Conjecture 2.5] (see also [40, Conjecture 4.4]). Indeed, it is equivalent to requiring
with
and
see (5.4), where
Conjecture 5.16 is trivially true when X is a curve, or in the toric case, if we restrict to toric norms. Indeed, in both cases χ will automatically be of finite type, and hence equal to
Remark 5.17.
In his pioneering work on K-stability for filtrations, G. Székelyhidi defined the Donaldson–Futaki invariant of a norm/filtration as
To the memory of Jean-Pierre Demailly, with admiration
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1600011
Award Identifier / Grant number: DMS-1900025
Award Identifier / Grant number: DMS-2154380
Funding statement: The first author was partially supported by the ANR grant GRACK. The second author was partially supported by NSF grants DMS-1600011, DMS-1900025, DMS-2154380, and the United States–Israel Binational Science Foundation.
A Log discrepancy and dual complexes
The goal of this appendix is to review the extension of the log discrepancy function on a projective klt pair from the set of divisorial valuations to the whole Berkovich space. We also describe this extension using dual complexes of test configurations.
A.1 The log discrepancy function on valuations
First let
to
Lemma A.1.
The log discrepancy function
Granted this result, we can define the extension as follows.
Definition A.2.
We define the log discrepancy
for all
In fact,
To prove Lemma A.1, and to get a more precise understanding of the log discrepancy function, we rely on [35], which gave a different way of extending the log discrepancy in the case when X is smooth and
By a log smooth pair
Let
Given
It follows that
Proof of Lemma A.1.
In view of [35, Lemma 5.7], the log discrepancy function
Proposition A.3.
When X is smooth, the log discrepancy function
To prove this, we need to recall the construction in [35].
Given
satisfying
We say that a log smooth pair
see [35, Theorem 4.9]. In particular,
Let us temporarily write
for any
for any
Proof of Proposition A.3.
We must prove that
by (c) and the lower semicontinuity of
The log discrepancy functions satisfy the following expected properties.
Proposition A.4.
Let
for any
for any projective birational morphism
Proof.
The equality in (b) holds on
A.2 Extension to the whole Berkovich space
Given a pair
Definition A.5.
For any sublc pair
the greatest lsc extension of
Note that a sublc pair
Theorem A.6.
For any sublc pair
for any
Proof.
Assume that
for all
we get
Thus
A.3 Valuations of finite log discrepancy
If
Definition A.7.
For any (not necessarily normal) projective variety X, we define the set
Clearly
Lemma A.8.
The set
Proof.
We may assume X is smooth. Applying Theorem A.6 to the klt pair
Remark A.9.
Similarly
A.4 Log discrepancy via snc test configurations
We refer to [20, Appendix A] for details on the following discussion. For any projective variety X, Gauss extension provides an embedding
onto the set of
Now assume that X is smooth, and consider an snc test configuration
for X, i.e. a test configuration dominating the trivial test configuration, such that
Let us write
see [20, Theorem A.1]. Thus
Theorem A.10.
If X is a smooth projective variety, then
on
and
Proof.
Set
Now
By [35] we have
Note that the equality
Acknowledgements
The authors would like to thank R. Berman, H. Blum, R. Dervan, E. Di Nezza, A. Ducros, V. Guedj, M. Hattori, E. Inoue, E. Legendre, C. Li, Yaxiong Liu, Y. Odaka, R. Reboulet, L. Sektnan, Z. Sjöström Dyrefelt, A. Zeriahi, and K. Zhang for fruitful discussions and useful comments.
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Articles in the same Issue
- Frontmatter
- A non-Archimedean approach to K-stability, II: Divisorial stability and openness
- Uniqueness of asymptotically conical higher codimension self-shrinkers and self-expanders
- A Schur’s theorem via a monotonicity and the expansion module
- On Waring’s problem for larger powers
- Tangent flows of Kähler metric flows
- Ramified covers of abelian varieties over torsion fields
- Filling minimality and Lipschitz-volume rigidity of convex bodies among integral current spaces
- Cartier smoothness in prismatic cohomology
Articles in the same Issue
- Frontmatter
- A non-Archimedean approach to K-stability, II: Divisorial stability and openness
- Uniqueness of asymptotically conical higher codimension self-shrinkers and self-expanders
- A Schur’s theorem via a monotonicity and the expansion module
- On Waring’s problem for larger powers
- Tangent flows of Kähler metric flows
- Ramified covers of abelian varieties over torsion fields
- Filling minimality and Lipschitz-volume rigidity of convex bodies among integral current spaces
- Cartier smoothness in prismatic cohomology
