Non-vanishing theorems for rank two vector bundles on threefolds

The paper investigates the non-vanishing of $H^1(\shE(n))$, where $\shE$ is a (normalized) rank two vector bundle over any smooth irreducible threefold $X$ with $\Pic(X) \cong \ZZ$. If $\epsilon$ is defined by the equality $\omega_X = \shO_X(\epsilon)$, and $\alpha$ is the least integer $t$ such that $H^0(\shE(t)) \ne 0$, then, for a non-stable $\shE$, $H^1(\shE(n))$ does not vanish at least between $\frac{\epsilon-c_1}{2}$ and $-\alpha-c_1-1$. The paper also shows that there are other non-vanishing intervals, whose endpoints depend on $\alpha$ and on the second Chern class of $\shE$. If $\shE$ is stable $H^1(\shE(n))$ does not vanish at least between $\frac{\epsilon-c_1}{2}$ 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.