Regularity and Cohomological Splitting Conditions for Vector Bundles on Multiprojective Spaces

Cohomological characterization of vector bundles on multiprojective spaces

We show that Horrock's criterion for the splitting of vector bundles on $\PP^n$ can be extended to vector bundles on multiprojective spaces and to smooth projective varieties with the weak CM property (see Definition 3.11). As a main tool we use the theory of $n$-blocks and Beilinson's type spectral sequences. Cohomological characterizations of vector bundles are also showed.

m-blocks collections and Castelnuovo-Mumford regularity in multiprojective spaces

The main goal of the paper is to generalize Castelnuovo-Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on $n$-dimensional smooth projective varieties $X$ with an $n$-block collection $\cB $ which generates the bounded derived category $\cD ^b({\cO}_X$-$mod)$. To this end, we use the theory of $n$-blocks and Beilinson type spectral sequence to define the notion of regularity of a coherent sheaf $F$ on $X$ with respect to the $n$-block collection $\cB $. We show that the basic formal properties of the Castelnuovo-Mumford regularity of coherent sheaves over proje…

Scrolls上的Castelnuovo-Mumford正则性与分裂准则

Castelnuovo-Mumford Regularity over Scrolls and Splitting Criteria

该研究提出适用于由射影空间上线丛和的射影化得到的scrolls的Castelnuovo-Mumford正则性概念，证明其为经典正则性的自然推广，并建立了向量丛的Horrocks型分裂准则。

向量丛在$\mathbb{P}^{n_1}\times\cdots\times\mathbb{P}^{n_s}$上的分裂的上同调准则

A Cohomological criterion for the splitting of vector bundles on $\mathbb{P}^{n_1}\times\cdots\times\mathbb{P}^{n_s}$

本文提出了一种判断向量丛在乘积射影空间上是否分裂的上同调准则，并推广了在对称乘积射影空间上的消失上同调准则。

$\mathbb{P}^{n_1}\times\cdots\times\mathbb{P}^{n_s}$ 上向量丛的上同调分裂条件

Cohomological splitting conditions of vector bundles on $\mathbb{P}^{n_1}\times\cdots\times\mathbb{P}^{n_s}$

该研究将 Ballico 和 Malaspina 关于多射影空间上向量丛正则性与分裂条件的结论推广到更一般情形。

A Few Splitting Criteria for Vector Bundles

We prove a few splitting criteria for vector bundles on a quadric hypersurface and Grassmannians. We give also some cohomological splitting conditions for rank 2 bundles on multiprojective spaces. The tools are monads and a Beilinson's type spectral sequence generalized by Costa and Miró-Roig.