Related papers: Elementary counterexamples to Kodaira vanishing in…
Let $X$ be a smooth projective variety over a perfect field $k$ of characteristic $p>0$, and $V$ be a vector bundle over $X$. It is well known that if $X$ is a curve and $V$ is not strongly semistable, then some Frobenius pullback…
We construct examples of threefolds with terminal singularities (resp. surfaces with canonical singularities) which are special in the sense of Campana, have a potentially dense set of integral points, admit a dense entire curve, have…
In this article, we get properties for singular (dual) Nakano semi-positivity and obtain singular type vanishing theorem involving $L^2$-subsheaves on weakly pseudoconvex manifolds by $L^2$-estimates and $L^2$-type Dolbeault isomorphisms.…
Let $X$ be a smooth projective curve of genus $g(X)\geq 1$ over an algebraically closed field $k$ of characteristic $p>0$, $\M^s_X(r,d)$ the moduli space of stable vector bundles of rank $r$ and degree $d$ on $X$. We study the Frobenius…
King's conjecture states that on every smooth complete toric variety $X$ there exists a strongly exceptional collection which generates the bounded derived category of $X$ and which consists of line bundles. We give a counterexample to this…
Given a sequence of genus $g\geq 2$ curves converging to a punctured Riemann surface with complete metric of constant Gaussian curvature $-1$. we prove that the Kodaira embedding using orthonormal basis of the Bergman space of sections of a…
This paper establishes a second vanishing theorem for formal local cohomology modules over Noetherian local rings. We introduce the \textit{formal dimension} invariant and characterize the vanishing of higher formal local cohomology in…
The purpose of this paper is to give two supplements for vanishing theorems: One is a relative version of the Kawamata-Viehweg-Nadel type vanishing theorem, which is obtained from an observation for the variation of the numerical dimension…
This paper proves the existence of potentials of the first and second kind of a Frobenius like structure in a frame which encompasses families of arrangements. Surprisingly the proof is based on the study of finite sets of vectors in a…
We discuss secondary (and higher) characteristic classes for algebraic vector bundles with trivial top Chern class. We show that if X is a smooth affine scheme of dimension d over a field k of finite 2-cohomological dimension (with char(k)…
A counterexample to the Jordan-H\"older property for semiorthogonal decompositions of derived categories of smooth projective varieties was constructed by B\"ohning, Graf von Bothmer and Sosna. In this short note we present a simpler…
The main theorem (2.2) consists in two characterizations of isomorphisms of factorial domains in terms of prime or primary rings elements, and unramified, flat or weakly injective affine schemes morphisms. In order to apply this theorem to…
The purpose of this note is to show that in characteristic two there exists a smooth Fano variety for which the square of the anti-canonical bundle violates Kodaira vanishing. This is used in turn to construct non-Cohen-Macaulay canonical…
The first part of the paper contains a detailed proof of M. Saito's generalization of the Kodaira vanishing theorem, following the original argument and with ample background, based on a lecture given at a Clay workshop on mixed Hodge…
Recently M. Mustata and V. Srinivas related a natural conjecture about the Frobenius action on the cohomology of the structure sheaf after reduction to characteristic $p > 0$ with another conjecture connecting multiplier ideals and test…
We study parabolic bundles on an algebraic curve in positive characteristic. Our motivation is to properly formulate Frobenius pull-backs of parabolic bundles in a way that extends various previous facts and arguments for the usual…
Let $X$ be a projective scheme over a field. We show that the vanishing cohomology of any sequence of coherent sheaves is closely related to vanishing under pullbacks by the Frobenius morphism. We also compare various definitions of ample…
This paper aims to develop a theory of projective and affine structures on higher-dimensional varieties in positive characteristic. This theory deals with Frobenius-projective and Frobenius-affine structures, which have been previously…
Let $f:X\rightarrow Z$ be a separable fibration of relative dimension 1 between smooth projective varieties over an algebraically closed field $k$ of positive characteristic. We prove the subadditivity of Kodaira dimension…
We prove an effective vanishing theorem for direct images of log pluricanonical bundles of projective semi-log canonical pairs. As an application, we obtain a semipositivity theorem for direct images of relative log pluricanonical bundles…