Related papers: Positivity of vector bundles and Dominance

Hartshorne in "Ample vector bundles" proved that $E$ is ample if and only if $\OOO_{P(E)}(1)$ is ample. Here we generalize this result to flag manifolds associated to a vector bundle $E$ on a complex manifold $X$: For a partition $a$ we…

We prove that certain vector bundles over surfaces are ample if they are so when restricted to divisors, certain numerical criteria hold, and they are semistable (with respect to $\det(E)$). This result is a higher-rank version of a theorem…

Given a covering f: X \to Y of projective manifolds, we consider the vector bundle E on Y given as the dual of f_*(\O_X) / \O_Y. This vector bundles often has positivity properties, e.g. E is ample when Y is projective space by a theorem of…

Let $X$ be a smooth projective variety of dimension $n$, and let $E$ be an ample vector bundle over $X$. We show that any non-zero Schur class of $E$, lying in the cohomology group of bidegree $(n-1, n-1)$, has a representative which is…

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…

We study the positivity of an Ulrich vector bundle defined with respect to a globally generated ample line bundle. First we prove a generalization of a Lopez theorem on the first Chern class and the bigness of an Ulrich bundle. Then, under…

In this paper, we study when positivity conditions of vector bundles are preserved by extension. We prove that an extension of a big (resp. pseudo-effective) line bundle by an ample (resp. a nef) vector bundle is big (resp.…

In this paper we prove a generalization of a theorem of Schneider, which gives a criterion for a projective surface over the complex numbers to have an ample cotangent bundle. After reviewing different notions of positivity, we introduce a…

Let $E$ be an ample vector bundle of rank $r$ on a projective variety $X$ with only log-terminal singularities. We consider the nefness of adjoint divisors $K_X+(t-r)det(E)$ when $t>=dim(X)$ and $t>r$. As a corollary, we classify pairs…

We show that if $E$ is an ample vector bundle of rank at least two with some curvature bound on $O_{P(E^*)}(1)$, then $E^*\otimes \det E$ is Kobayashi positive. The proof relies on comparing the curvature of $(\det E^*)^k$ and $S^kE$ for…

We study Seshadri constants of certain ample vector bundles on projective varieties. Our main motivation is the following question: Under what conditions are the Seshadri constants of ample vector bundles at least 1 at all points of the…

Let $X$ be a $d$ dimensional projective manifold, $E$ be an ample vector bundle on $X$ and $0\le \lambda_N\le \lambda_{N-1} \le \cdots \le \lambda_1 \le \operatorname{rank}(E)$ be a partition of $d-2$. We prove that the Schur class…

Let $\sE$ be an ample rank $r$ bundle on a smooth toric projective surface, $S$, whose topological Euler characteristic is $e(S)$. In this article, we prove a number of surprisingly strong lower bounds for $c_1(\sE)^2$ and $c_2(\sE)$. We…

We study the following question: Given a vector bundle on a projective variety $X$ such that the restriction of $E$ to every closed curve $C \,\subset\, X$ is ample, under what conditions $E$ is ample? We first consider the case of an…

It is well known that positivity properties of the curvature of a vector bundle have implications on the algebro-geometric properties of the bundle, such as numerical positivity, vanishing of higher cohomology leading to existence of global…

We examine the relationship between positivity of a vector bundle E and the problem of $L^2$ extension of holomorphic E-valued forms of top degree. In particular, we show by example that Griffiths positivity is not enough. Though the…

Let $X$ be the special fiber of a unitary Shimura variety of hyperspecial level at a prime $p$ inert in the totally real field $F$. Let $Y\to X$ be the associated flag space. For every $L$-dominant weight $\lambda$, let…

Combinatorial ideas are developed in this article to study Chern numbers on ample and numerically effective vector bundles. An effective lower bound for Chern numbers of ample vector bundles is established, which makes some progress towards…

We show that an equivariant vector bundle on a complete toric variety is nef or ample if and only if its restriction to every invariant curve is nef or ample, respectively. Furthermore, we show that nef toric vector bundles have a…

We show that a nef line bundle on a proper scheme over an excellent base is semiample if and only if it is semiample after restricting to characteristic zero and to positive characteristic. In the process of the proof, we provide a…