Non-vanishing theorems for rank two vector bundles on threefolds

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 $Pic(X)\cong \text{ZZ}$. If $ϵ$ is the integer defined by the equality ${\omega }_X=O_X(ϵ)$, and $\alpha$ is the least integer $t$ such that $H^0(E(t))\mathrm{\ne }0$, then, for a non-stable $E$ ($\alpha \le 0$) the first cohomology module does not vanish at least between the endpoints $\frac{ϵ-c_1}{2}$ 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 $\frac{ϵ-c_1}{2}$ and $\alpha -2$. The paper considers also the case of a threefold $X$ with $Pic(X)\mathrm{\ne }\text{ZZ}$ but $Num(X)\cong \text{ZZ}$ and gives similar non-vanishing results.