Non-vanishing theorems for rank two vector bundles on threefolds

TL;DR

This paper establishes new non-vanishing theorems for the first cohomology of rank two vector bundles on threefolds, detailing specific intervals where cohomology does not vanish based on stability and Chern classes.

Contribution

It provides explicit non-vanishing intervals for $H^1(E(n))$ depending on stability, Chern classes, and the Picard group structure of the threefold, extending previous results.

Findings

Non-vanishing intervals depend on stability and Chern classes.

Results apply to threefolds with Picard group isomorphic to integers.

Similar non-vanishing results hold for threefolds with different Picard group structures.

Abstract

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 equality $\omega_X = O_X(\epsilon)$, and $\alpha$ is the least integer $t$ such that $H^0(E(t)) \ne 0$, then, for a non-stable $E$ ($\alpha \le 0$) the first cohomology module does not vanish at least between the endpoints $\frac{\epsilon-c_1}{2}$ and $-\alpha-c_1-1$. The paper also shows that there are other non-vanishing intervals, whose endpoints depend on $\alpha$ and also on the second Chern class $c_2$ of $E$. If $E$ is stable the first cohomology module does not vanish at least between the endpoints $\frac{\epsilon-c_1}{2}$ and $\alpha-2$. The paper considers also the case of a threefold $X$ with $Pic(X) \ne \ZZ$ but $Num(X) \cong \ZZ$ and gives similar non-vanishing results.