Non-vanishing Theorems for Rank Two Vector Bundles on Threefolds

Abstract

The paper investigates the non-vanishing of $H^{1}\left(\varepsilon\left(n\right)\right)$, where $\varepsilon$ is a (normalized) rank two vector bundle over any smooth irreducible threefold X with $PIC\left(X\right)\cong\mathbb{Z}$. If $\epsilon$ is defined by the equality $\omega_{X}=\mathcal{O}_{X}\left(\epsilon\right)$ and $\alpha$ is the least integer t such that $H^{t}\left(\varepsilon\left(t\right)\right)\neq0$, then, for a non-stable $\varepsilon$, $H^{1}\left(\varepsilon\left(n\right)\right)$ 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 a and on the second Chem class of $\varepsilon$. If $\varepsilon$ is stable $H^{1}\left(\varepsilon\left(n\right)\right)$ 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 $PIC\left(X\right)\neq\mathbb{Z}$ but $Num\cong\mathbb{Z}$ and gives similar non-vanishing results.