Vanishing theorems for products of exterior and symmetric powers

Author name not available (Why is that?)

Abstract: For ample vector bundles

E

over compact complex varieties

X

and a Schur functor

S_{I}

corresponding to an arbitrary partition

I

of the integer

| I |

, one would like to know the optimal vanishing theorem for the cohomology groups

H^{p, q} (X, S_{I} (E))

, depending on the rank of

E

and the dimension

n

of

X

. Three years ago (Nov. 1995), in an unpublished paper one of us (W.N.) proved a vanishing theorem for the situation where the partition

I

is a hook. Here we give a simpler proof of this theorem. We also treat the same problem under weaker positivity assumptions, in particular under the hypothesis of ample

L a m b d a^{m} E

with

m i n N^{*}

. In this case we also need some bound on the weight

| I |

of the partition. Moreover, we prove that the same vanishing condition applies for

H^{q, p} (X, S_{I} (E))

, with

p, q

interchanged.

Mathematics Subject Classification ID

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