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.
