Related papers: Globally $F$-regular and log Fano varieties

In this paper, we prove that a weak form of the Akizuki-Nakano vanishing theorem holds on globally $F$-split 3-folds. Making use of this vanishing theorem, we study deformations of globally $F$-split Fano 3-folds and the Kodaira vanishing…

We extend the notion of Frobenius Betti numbers and F-splitting ratio to large classes of finitely generated modules over rings of prime characteristic, which are not assumed to be local. We also prove that the strong F-regularity of a pair…

We prove that if $X$ is a smooth Fano threefold and $L$ is an ample $\mathbb{Q}$-divisor such that $(X,L)$ is K-polystable, then the automorphism group $\operatorname{Aut}(X)$ is reductive. This verifies the reductivity statement predicted…

In this article we study the Honda-Tate theory for log abelian varieties over an fs log point $S=(\mathrm{Spec}(\mathbf{k}),M_S)$ for $\mathbf{k}=\mathbb{F}_q$ a finite field, generalizing the classical Honda-Tate theory for abelian…

We prove some criteria for uniform K-stability of log Fano pairs. In particular, we show that uniform K-stability is equivalent to $\beta$-invariant having a positive lower bound. Then we study the relation between optimal destabilization…

The Green-Griffiths-Lang conjecture stipulates that for every projective variety X of general type over C, there exists a proper algebraic subvariety of X containing all non constant entire curves f : C $\rightarrow$ X. Using the formalism…

We prove the bigness of the Chow-Mumford line bundle associated to a $\mathbb{Q}$-Gorenstein family of log Fano varieties of maximal variation with uniformly K-stable general geometric fibers. This result generalizes a recent theorem of…

We show that G-equivariant K-semistability (resp. G-equivariant K-polystability) implies K-semistability (resp. K-polystability) for log Fano pairs when G is a finite group.

We study Fano 3-folds with Fano index 2: that is, 3-folds X with rank Pic(X) = 1, Q-factorial terminal singularities and -K_X = 2A for an ample Weil divisor A. We give a first classification of all possible Hilbert series of such polarised…

Let $p$ be a prime integer and $F$ a field of characteristic 0. Let $X$ be the {\em norm variety} of a symbol in the Galois cohomology group $H^{n+1}(F,\mu_p^{\otimes n})$ (for some $n\geq1$), constructed in the proof of the Bloch-Kato…

We construct a generalization of the Hasse invariant for certain unitary Shimura varieties of PEL type whose vanishing locus is the complement of the so-called \mu-ordinary locus. We show that the \mu-ordinary locus of those varieties is…

For a normal projective variety $X$, the $\bf Q$-factoriality defect $\sigma(X)$ is defined to be the rank of the quotient of the group of Weil divisors by the subgroup of Cartier ones. We prove a slight improvement of a topological formula…

We prove that Schubert varieties are globally F-regular in the sense of Karen Smith. We apply this result to the category of equivariant and holonomic D-modules on flag varieties in positive characteristic. Here recent results of Blickle…

We describe a natural basis of the Cartier class group of an arbitrary Schubert variety $X_{w,P}$ in a flag variety $G/P$ of general Lie type. We then characterise when the Schubert variety is factorial/Fano, along with an explicit formula…

We relate the global log canonical threshold of a variety with torus action to the global log canonical threshold of its quotient. We apply this to certain Fano varieties and use Tian's criterion to prove the existence of Kahler-Einstein…

A projective log variety (X, D) is called "a log Fano manifold" if X is smooth and if D is a reduced simple normal crossing divisor on X with -(K_X+D) ample. The n-dimensional log Fano manifolds (X, D) with nonzero D are classified in this…

We give counterexamples to the Kawamata-Viehweg vanishing theorem on ruled surfaces in positive characteristic, and prove that if there is a counterexample to the Kawamata-Viehweg vanishing theorem on a geometrically ruled surface f:X-->C,…

We investigate the relationship between the Fano type property on fibers over a Zariski dense subset and the global Fano type property. We establish the invariance of N\'eron-Severi spaces, nef cones, effective cones, movable cones, and…

Let $X$ be a smooth projective variety over an algebraically closed field $\mathbb{K}$ with arbitrary characteristic. Suppose $L$ is an ample and globally generated line bundle. By Castelnuovo--Mumford regularity, we show that $K_X \otimes…

Kawamata proposed a conjecture predicting that every nef and big line bundle on a smooth projective variety with trivial first Chern class has nontrivial global sections. We verify this conjecture for several cases, including (i) all…