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Non-vanishing Theorems for Rank Two Vector Bundles on Threefolds
Ballico, Edoardo
Valabrega, Paolo
Valenzano, Mario
2011
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.
Series
Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics
43 (2011)
Publisher
EUT Edizioni Università di Trieste
Source
Edoardo Ballico, Paolo Valabrega and Mario Valenzano, "Non-vanishing Theorems for Rank Two Vector Bundles on Threefolds", in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 43 (2011), pp. 11–30
Languages
en
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