Regularity and cohomological splitting conditions for vector bundles on multiprojective spaces (Q408506)

The authors consider a general product of two projective spaces \(X=\mathbb P^n\times \mathbb P^m\) and study the cohomology of vector bundles over \(X\). They start with a new definition of regularity, for coherent sheaves on \(X\), which modifies a previous definition given by \textit{J. W. Hoffman} and \textit{H. H. Wang} [Adv. Geom. 4, No. 4, 513--536 (2004; Zbl 1063.13001)]. In details, \(F\) is \((p,p')\)-regular if \(H^i(F(p,p')\otimes \mathcal O (j,k))=0\) for \(-n\leq j\leq 0\), \(-m\leq k\leq 0\) and \(j+k=-i\). The authors prove several properties of regular sheaves on the product \(X\), most of which are similar to the properties of Castelnuovo-Mumford regular sheaves on projective spaces . In the spirit of the new definition, the authors extend Horrocks splitting criterion for vector bundles. Indeed, they show that a rank \(r\) bundle \(E\) on \(X\) splits in a sum of line bundles \(E=\oplus^r\mathcal O(t_s,t_s)\) if and only if for \(i=1,\dots,m+n-1\) and for all \(t\), the cohomology groups \(H^i(E(t,t)\otimes \mathcal O (j,k))\) vanish whenever \(-n\leq j\leq 0\), \(-m\leq k\leq 0\) and \(j+k=-i\).