On positivity of the CM line bundle on K-moduli spaces

Xu, Chenyang; Zhuang, Ziquan

doi:10.4007/annals.2020.192.3.7

Cited by 40 publications

(82 citation statements)

References 48 publications

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“…If the log version of the Odaka–Spotti–Sun conjecture holds, as

β

is perturbed the variation of the positivity of the CM line bundle over families would detect wall‐crossing phenomena in the compact moduli space. Remark The ampleness of the log CM line bundle has been recently established in the pre‐print [58]. …”

Section: Proofs Of the Main Theoremsmentioning

confidence: 99%

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Applications of the moduli continuity method to log K‐stable pairs

Gallardo

Martinez-Garcia

Spotti

2020

J. London Math. Soc.

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The 'moduli continuity method' permits an explicit algebraisation of the Gromov-Hausdorff compactification of Kähler-Einstein metrics on Fano manifolds in some fundamental examples. In this paper, we apply such method in the 'log setting' to describe explicitly some compact moduli spaces of K-polystable log Fano pairs. We focus on situations when the angle of singularities is perturbed in an interval sufficiently close to one, by considering constructions arising from Geometric Invariant Theory. More precisely, we discuss the cases of pairs given by cubic surfaces with anticanonical sections, and of projective space with non-Fano hypersurfaces, and we show ampleness of the CM line bundle on their good moduli space (in the sense of Alper). Finally, we introduce a conjecture relating K-stability (and degenerations) of log pairs formed by a fixed Fano variety and pluri-anticanonical sections to certain natural GIT quotients. GIT d of degree d hypersurfaces in P n. The first statement of the above result is a special case of the following natural expectation, at least when the automorphism group has no non-trivial characters. Conjecture 1.3. Let X be a K-polystable Fano variety. Then for any sufficiently large and divisible l ∈ N, there exists β 0 = β 0 (l, X) such that for all D ∈ | − lK X | and β ∈ (β 0 , 1), the pair (X, (1 − β)D) is log Kpolystable if and only if [D] ∈ P(H 0 (−lK X)) is GIT-polystable for the natural representation of Aut(X) on H 0 (−lK X). Moreover, the Gromov-Hausdorff compactification M K X,l,β of the above pairs (X, (1 − β)D) is homeomorphic to the GIT quotient P(H 0 (−lK X)) ss //Aut(X). Note that the automorphism group Aut(X) of a K-polystable Fano is always reductive. If X is smooth or a Gromov-Hausdorff degeneration, this follows from by Matsushima's obstruction [Mat57] and Chen-Donaldson-Sun [CDS15]. Otherwise, this has been very recently established in [ABHLX20] via purely algebro-geometric techniques Once its different steps are established, the application of the moduli continuity method (the proof of theorems 1.1 and 1.2) is just a couple of paragraphs. However, in order to keep the reader focused on the main goal of the paper, we recall here what the different steps of the moduli continuity method are, noting that each of them requires involved technical proofs to accomplish them. Suppose that we have a Gromov-Hausdorff

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“…If the log version of the Odaka–Spotti–Sun conjecture holds, as

β

Section: Proofs Of the Main Theoremsmentioning

confidence: 99%

“…Remark 4.9. The ampleness of the log CM line bundle has been recently established in the pre-print [58].…”

Section: Wall-crossingmentioning

confidence: 99%

Applications of the moduli continuity method to log K‐stable pairs

Gallardo

Martinez-Garcia

Spotti

2020

J. London Math. Soc.

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“…From this powerful theory, one can construct a satisfactory moduli space of K-stable Fano varieties, the so-called K-moduli space which is proper as proved recently in [32]. There are many works along these lines, see [7], [15], [6], [41], [1], [4], [42], etc. The valuative criterion for K-stability of Fano varieties has played an essential role in all of these developments.…”

Section: Introductionmentioning

confidence: 98%

Openness of uniformly valuative stability on the Kähler cone of projective manifolds

Liu¹

2021

Preprint

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Assume that a projective variety with a polarization is uniformly valuatively stable (stronger than uniformly valuatively stable given by Dervan and Legendre in [19]), then the projective variety with any polarization sufficiently close to the original polarization is uniformly valuatively stable. We also define valuative stability for the transcendental Kähler classes. Our openness result can be extended to the Kähler cone of projective manifolds.

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“…The Chow-Mumford (CM) line bundle is an ample line bundle on this moduli space. We refer to the introductions of [CP21] and [XZ20], to the survey [?] and to the recent groundbreaking paper [?]…”

Section: Introductionmentioning

confidence: 99%

“…The novelty of this new proof is similar as for the proof of Theorem 1.2, which was explained after the statement of Theorem 1.2. We also note that Theorem 1.3 is used to prove the positivity of the Chow-Mumford line bundle under convenient assumptions, see [CP21,Pos19,XZ20].…”

Section: Introductionmentioning

confidence: 99%

A note on families of K-semistable log-Fano pairs

Codogni¹,

Patakfalvi²

2021

Preprint

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In this short note, we give an alternative proof of the semipositivity of the Chow-Mumford line bundle for families of K-semistable log-Fano pairs, and of the nefness threeshold for the log-anti-canonical line bundle on families of K-stable log Fano pairs. We also prove a bound on the multiplicity of fibers for families of K-semistable log Fano varieties, which to the best of our knowledge is new.

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On positivity of the CM line bundle on K-moduli spaces

Cited by 40 publications

References 48 publications

Applications of the moduli continuity method to log K‐stable pairs

Applications of the moduli continuity method to log K‐stable pairs

Openness of uniformly valuative stability on the Kähler cone of projective manifolds

A note on families of K-semistable log-Fano pairs

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