Invariants of Fano Varieties in Families

Gounelas, Frank; Javanpeykar, Ariyan

doi:10.17323/1609-4514-2018-18-2-305-319

Cited by 10 publications

(13 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…When q = p, note that p ∤ c 1 ⋅ NS(S), the argument in Step 2 actually shows that there exists an element x in (33) or (34) such that p ∤ c 1 ⋅ x. This implies that U p is non-empty and hence proves the assertion.…”

Section: 4mentioning

confidence: 72%

See 1 more Smart Citation

Supersingular Irreducible Symplectic Varieties

2021

Rationality of Varieties

View full text Add to dashboard Cite

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

show abstract

Section: 4mentioning

confidence: 72%

“…Theorem 2.8 (cf. [34,Theorem 1.2]). Let X be smooth projective variety over k. If X is rationally chain connected, then the first Chern class map induces an isomorphism Pic(X)⊗Q ≅ H 2 ét (X, Q (1)) for all ≠ p. In particular, ρ(X) = b 2 (X) and X is 2 nd -Shioda supersingular.…”

Section: Generalities On the Notion Of Supersingularitymentioning

confidence: 99%

Supersingular Irreducible Symplectic Varieties

2021

Rationality of Varieties

View full text Add to dashboard Cite

show abstract

“…We prove two cases of this for specializations to positive characteristic. The first case uses "prime-to-p decompositions of the diagonal" and results of Totaro and Gounelas-Javanpeykar on de Rham cohomology and Picard groups for specializations of Fano manifolds, [Tot16], [GJ18]. This version applies in the ramified case and bounds the "bad primes" as divisors of Tor(K,…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…By [GJ18], the closed fiber of X R has cyclic Picard group. Since the closed fiber is separably uniruled, by [Tia15, Theorem 5, Corollary 9], the closed fiber is also freely rationally connected, and the closed fiber is even separably rationally connected when it is a complete intersection.…”

Section: Free Lines On Complete Intersectionsmentioning

confidence: 99%

Weak approximation for Fano complete intersections in positive characteristic

Starr¹,

Tian²,

Zong³

2018

Preprint

View full text Add to dashboard Cite

For a smooth curve B over a field k = k with char(k) = p, for every B-flat complete intersection X B in B × Spec k P n k of type (d 1 , . . . , dc), we prove weak approximation of O B,b -points of X B by k(B)-points at all places b of (strong) potentially good reduction, if the Fano index is ≥ 2 and if p > max(d 1 , . . . , dc). Assuming this inequality, which is nearly sharp, such complete intersections are separably uniruled by lines, and even separably rationally connected. This also applies to specializations of complex Fano manifolds with Picard rank 1 and Fano index 1 away from "bad primes".

show abstract

“…Moreover, by Chen-Donaldson-Sun [12], we know that W is deformation equivalent to P n . Since the Fano index is constant along smooth deformations [26,Proposition 6.2], the Fano index of W is n + 1. Hence, W is biholomorphic to P n by Kobayashi-Ochiai [31], and the claims follow.…”

Section: Proofs Of the Main Theoremsmentioning

confidence: 99%

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

Gallardo

Martinez-Garcia

Spotti

2020

J. London Math. Soc.

View full text Add to dashboard Cite

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

show abstract

Invariants of Fano Varieties in Families

Cited by 10 publications

References 0 publications

Supersingular Irreducible Symplectic Varieties

Supersingular Irreducible Symplectic Varieties

Weak approximation for Fano complete intersections in positive characteristic

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

Contact Info

Product

Resources

About