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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 integerdefined by the equality $\omega_X = O_X(\epsilon)$, and $\alpha$ is the leastinteger $t$ 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 theendpoints $\frac{\epsilon-c_1}{2}$ and $-\alpha-c_1-1$. The paper also showsthat 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 thefirst 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 athreefold $X$ with $Pic(X) \ne \ZZ$ but $Num(X) \cong \ZZ$ and gives similarnon-vanishing results.

Non-vanishing theorems for rank two vector bundles on threefolds