Related papers: Remarks on Ramanujam-Kawamata-Viehweg Vanishing Th…

We prove a Kawamata-Viehweg vanishing theorem on a normal compact Kahler space X: if L is a nef line bundle with numerical dimension at least equal to 2, then the q-th cohomology group of K_X+L vanishes for q at least equal to the dimension…

The paper investigates the non-vanishing of $H^1(E(n))$, where $E$ is a (normalized) rank two vector bundle over any smooth irreducible threefold $X$ of degree $d$ such that $Pic(X) \cong \ZZ$. If $\epsilon$ is the integer defined by the…

In this paper, we study the Nakano-positivity and dual-Nakano-positivity of certain adjoint vector bundles associated to ample vector bundles. As applications, we get new vanishing theorems about ample vector bundles. For example, we prove…

Let $X$ be a submanifold of dimension $n$ of the complex projective space $\mathbb P^N$ ($n<N$), and let $E$ be a vector bundle of rank two on $X$ . If $n\geq\frac{N+3}{2}\geq 4$ we prove a geometric criterion for the existence of an…

We consider smooth moduli spaces of semistable vector bundles of fixed rank and determinant on a compact Riemann surface $X$ of genus at least $3$. The choice of a Poincar\'e bundle for such a moduli space $M$ induces an isomorphism between…

Let $X$ be a smooth complete curve, and let $Bun_n$ be the moduli stack of rank $n$ vector bundles on $X$. Let $E$ be a local system on $X$. In a recent paper of E.Frenkel, K.Vilonen and the author, it was shown that the vanishing of a…

By proving an integral formula of the curvature tensor of $E\ts \det E$, we observe that the curvature tensor of $E\ts \det E$ is very similar to that of a line bundle and obtain certain new Kodaira-Akizuki-Nakano type vanishing theorems…

According to Miyaoka, a vector bundle E on a smooth projective curve is semistable if and only if a certain numerical class in the projectivized bundle PE is nef. We establish a similar criterion for the semistability of Higgs bundles:…

Let $\mathcal{E}$ be an ample vector bundle of rank $r\geq 2$ on a smooth complex projective variety $X$ of dimension $n$. The aim of this paper is to describe the structure of pairs $(X,\mathcal{E})$ as above whose adjoint bundles…

In this work we deal with vector bundles of rank two on a Fano manifold $X$ with $b_2=b_4=1$. We study the nef and pseudoeffective cones of the corresponding projectivizations and how these cones are related to the decomposability of the…

We consider a complete nonsingular variety $X$ over $\bC$, having a normal crossing divisor $D$ such that the associated logarithmic tangent bundle is generated by its global sections. We show that $H^i\big(X, L^{-1} \otimes \Omega_X^j(\log…

In this paper we study smooth complex projective varieties $X$ containing a Grassmannian of lines $G(1,r)$ which appears as the zero locus of a section of a rank two nef vector bundle $E$. Among other things we prove that the bundle $E$…

We introduce a notion of singular hermitian metrics (s.h.m.) for holomorphic vector bundles and define positivity in view of $L^2$-estimates. Associated with a suitably positive s.h.m. there is a (coherent) sheaf 0-th kernel of a certain…

The paper investigates vanishing conditions on the first cohomology module of a normalized rank 2 vector bundle E on P^3 which force E to split, and finds therefore strategic levels of non-vanishing for a non-split bundle. The present…

A complex manifold $X$ of dimension $n$ together with an ample vector bundle $E$ on it will be called a {\sf generalized polarized variety}. The adjoint bundle of the pair $(X,E)$ is the line bundle $K_X + det(E)$. We study the positivity…

Let $E$ be a vector bundle of rank $r$ on a smooth complex projective variety $X$. In this article, we compute the nef and pseudoeffective cones of divisors in the Grassmann bundle $Gr_X(k,E)$ parametrizing $k$-dimensional subspaces of the…

We show that all filtrable bundles on a Hopf surface $X$ must have jumps and we prove the existence of filtrable stable bundles on $X$ with any value of $c_2>0$. On a somewhat opposite direction, for each integer $r\ge 2$ we prove the…

Let $E$ be a Hermitian vector bundle over a complete K\"{a}hler manifold $(X,\omega)$, $\dim_{\mathbb{C}}X=n$, with a $d$(bounded) K\"{a}hler form $\omega$, $d_{A}$ be a Hermitian connection on $E$. The goal of this article is to study the…

The purpose of this paper is to study minimal monads associated to a rank two vector bundle $\mathcal E$ on $\mathbb P^n$. In particular, we study situations where $\mathcal E$ has $H^i_*(\mathcal E) =0$ for $1<i<n-1$, except for one pair…

This article contains a new argument which proves vanishing of the first cohomology for negative vector bundles over a complex projective variety if the rank of the bundle is smaller than the dimension of the base. Similar argument is…