scholarly journals ON THE SHARPNESS OF TIAN’S CRITERION FOR K-STABILITY

2020 ◽  
pp. 1-33
Author(s):  
YUCHEN LIU ◽  
ZIQUAN ZHUANG
Keyword(s):  
Fano Variety ◽  
Fano Varieties

Abstract Tian’s criterion for K-stability states that a Fano variety of dimension n whose alpha invariant is greater than ${n}{/(n+1)}$ is K-stable. We show that this criterion is sharp by constructing n-dimensional singular Fano varieties with alpha invariants ${n}{/(n+1)}$ that are not K-polystable for sufficiently large n. We also construct K-unstable Fano varieties with alpha invariants ${(n-1)}{/n}$ .

scholarly journals Multiplicative Chow–Künneth decompositions and varieties of cohomological K3 type

2021 ◽  
Author(s):  
Lie Fu ◽  
Robert Laterveer ◽  
Charles Vial
Keyword(s):  
Projective Variety ◽  
Arbitrary Dimension ◽  
Smooth Projective Variety ◽  
The Other ◽  
Fano Variety ◽  
Fano Varieties ◽  
Intersection Product ◽  
Canonical Class ◽  
Main Input

AbstractGiven a smooth projective variety, a Chow–Künneth decomposition is called multiplicative if it is compatible with the intersection product. Following works of Beauville and Voisin, Shen and Vial conjectured that hyper-Kähler varieties admit a multiplicative Chow–Künneth decomposition. In this paper, based on the mysterious link between Fano varieties with cohomology of K3 type and hyper-Kähler varieties, we ask whether Fano varieties with cohomology of K3 type also admit a multiplicative Chow–Künneth decomposition, and provide evidence by establishing their existence for cubic fourfolds and Küchle fourfolds of type c7. The main input in the cubic hypersurface case is the Franchetta property for the square of the Fano variety of lines; this was established in our earlier work in the fourfold case and is generalized here to arbitrary dimension. On the other end of the spectrum, we also give evidence that varieties with ample canonical class and with cohomology of K3 type might admit a multiplicative Chow–Künneth decomposition, by establishing this for two families of Todorov surfaces.

scholarly journals K-stability of birationally superrigid Fano varieties

2019 ◽  
Vol 155 (9) ◽  
pp. 1845-1852 ◽  
Author(s):  
Charlie Stibitz ◽  
Ziquan Zhuang
Keyword(s):  
Moduli Space ◽  
Fano Variety ◽  
Fano Varieties

We prove that every birationally superrigid Fano variety whose alpha invariant is greater than (respectively no smaller than) $\frac{1}{2}$ is K-stable (respectively K-semistable). We also prove that the alpha invariant of a birationally superrigid Fano variety of dimension $n$ is at least $1/(n+1)$ (under mild assumptions) and that the moduli space (if it exists) of birationally superrigid Fano varieties is separated.

scholarly journals Smoothings of Fano Varieties With Normal Crossing Singularities

2015 ◽  
Vol 58 (3) ◽  
pp. 787-806
Author(s):  
Nikolaos Tziolas
Keyword(s):  
Characteristic Zero ◽  
Singular Locus ◽  
Total Space ◽  
Fano Variety ◽  
Closed Field ◽  
Fano Varieties ◽  
Algebraically Closed Field ◽  
Normal Crossing

AbstractThis paper obtains criteria for a Fano variety X defined over an algebraically closed field of characteristic zero with normal crossing singularities to be smoothable. In particular, we show that X is smoothable by a flat deformation X → Δ with smooth total space X if and only if where D is the singular locus of X.

scholarly journals The volume of singular Kähler–Einstein Fano varieties

2018 ◽  
Vol 154 (6) ◽  
pp. 1131-1158 ◽  
Author(s):  
Yuchen Liu
Keyword(s):  
Recent Result ◽  
Type Inequality ◽  
Upper Bounds ◽  
Fano Variety ◽  
Fano Varieties ◽  
Fano Manifold ◽  
Volume Function ◽  
Affine Cone ◽  
Quotient Singularities

We show that the anti-canonical volume of an $n$-dimensional Kähler–Einstein $\mathbb{Q}$-Fano variety is bounded from above by certain invariants of the local singularities, namely $\operatorname{lct}^{n}\cdot \operatorname{mult}$ for ideals and the normalized volume function for real valuations. This refines a recent result by Fujita. As an application, we get sharp volume upper bounds for Kähler–Einstein Fano varieties with quotient singularities. Based on very recent results by Li and the author, we show that a Fano manifold is K-semistable if and only if a de Fernex–Ein–Mustaţă type inequality holds on its affine cone.

scholarly journals Toric Degenerations and Laurent Polynomials Related to Givental's Landau–Ginzburg Models

2016 ◽  
Vol 68 (4) ◽  
pp. 784-815 ◽  
Author(s):  
Charles F. Doran ◽  
Andrew Harder
Keyword(s):  
Toric Varieties ◽  
Fano Variety ◽  
Complete Intersections ◽  
Flag Manifolds ◽  
Fano Varieties ◽  
Laurent Polynomials ◽  
Toric Degenerations ◽  
Geometric Transitions ◽  
Weak Fano

AbstractFor an appropriate class of Fano complete intersections in toric varieties, we prove that there is a concrete relationship between degenerations to speciûc toric subvarieties and expressions for Givental's Landau–Ginzburg models as Laurent polynomials. As a result, we show that Fano varieties presented as complete intersections in partial flag manifolds admit degenerations to Gorenstein toric weak Fano varieties, and their Givental Landau–Ginzburg models can be expressed as corresponding Laurent polynomials.We also use this to show that all of the Laurent polynomials obtained by Coates, Kasprzyk and Prince by the so–called Przyjalkowski method correspond to toric degenerations of the corresponding Fano variety. We discuss applications to geometric transitions of Calabi–Yau varieties.

scholarly journals Delta-invariants for Fano varieties with large automorphism groups

2020 ◽  
Vol 31 (10) ◽  
pp. 2050077
Author(s):  
Aleksei Golota
Keyword(s):  
Finite Group ◽  
Automorphism Groups ◽  
Fano Variety ◽  
Fano Varieties ◽  
Connected Subgroup ◽  
Closed Connected Subgroup ◽  
A Finite Group ◽  
Large Groups

For a variety [Formula: see text], a big [Formula: see text]-divisor [Formula: see text] and a closed connected subgroup [Formula: see text] we define a [Formula: see text]-invariant version of the [Formula: see text]-threshold. We prove that for a Fano variety [Formula: see text] and a connected subgroup [Formula: see text] this invariant characterizes [Formula: see text]-equivariant uniform [Formula: see text]-stability. We also use this invariant to investigate [Formula: see text]-equivariant [Formula: see text]-stability of some Fano varieties with large groups of symmetries, including spherical Fano varieties. We also consider the case of [Formula: see text] being a finite group.

scholarly journals On properness of K-moduli spaces and optimal degenerations of Fano varieties

2021 ◽  
Vol 27 (4) ◽  
Author(s):  
Harold Blum ◽  
Daniel Halpern-Leistner ◽  
Yuchen Liu ◽  
Chenyang Xu
Keyword(s):  
Moduli Spaces ◽  
Fano Varieties

scholarly journals Betti numbers and pseudoeffective cones in 2-Fano varieties

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Giosuè Emanuele Muratore
Keyword(s):  
Large Class ◽  
Betti Numbers ◽  
Fano Varieties ◽  
Higher Dimensional ◽  
Special Case ◽  
Answering Questions

Abstract The 2-Fano varieties, defined by De Jong and Starr, satisfy some higher-dimensional analogous properties of Fano varieties. We consider (weak) k-Fano varieties and conjecture the polyhedrality of the cone of pseudoeffective k-cycles for those varieties, in analogy with the case k = 1. Then we calculate some Betti numbers of a large class of k-Fano varieties to prove some special case of the conjecture. In particular, the conjecture is true for all 2-Fano varieties of index at least n − 2, and we complete the classification of weak 2-Fano varieties answering Questions 39 and 41 in [2].

scholarly journals Geometry and topology of the space of Kähler metrics on singular varieties

2018 ◽  
Vol 154 (8) ◽  
pp. 1593-1632 ◽  
Author(s):  
Eleonora Di Nezza ◽  
Vincent Guedj
Keyword(s):  
Einstein Metrics ◽  
Normal Space ◽  
Fano Varieties ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Metric Properties ◽  
Singular Varieties ◽  
Underlying Space ◽  
And Topology ◽  
Kähler Einstein Metrics

Let $Y$ be a compact Kähler normal space and let $\unicode[STIX]{x1D6FC}\in H_{\mathit{BC}}^{1,1}(Y)$ be a Kähler class. We study metric properties of the space ${\mathcal{H}}_{\unicode[STIX]{x1D6FC}}$ of Kähler metrics in $\unicode[STIX]{x1D6FC}$ using Mabuchi geodesics. We extend several results of Calabi, Chen, and Darvas, previously established when the underlying space is smooth. As an application, we analytically characterize the existence of Kähler–Einstein metrics on $\mathbb{Q}$-Fano varieties, generalizing a result of Tian, and illustrate these concepts in the case of toric varieties.

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