Vanishing theorems for products of exterior and symmetric powers

Original · EN

For ample vector bundles E over compact complex varieties X and a Schur functor Sᵢ corresponding to an arbitrary partition I of the integer |I|, one would like to know the optimal vanishing theorem for the cohomology groups Hᵖ,q(X, Sᵢ(E)), depending on the rank of E and the dimension n of X. Three years ago (Nov. 1995), in an unpublished paper one of us (W.N.) proved a vanishing theorem for the situation where the partition I is a hook.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 Λᵐ E with m∈. 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 Hq,ᵖ(X, Sᵢ(E)), with p,q interchanged.