Mixed Lefschetz Theorems and Hodge-Riemann Bilinear Relations

E Cattani, University of Massachusetts - Amherst

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This is the pre-published version harvested from ArXiv. The published version is located at http://imrn.oxfordjournals.org/content/2008/rnn025.refs

Abstract

The Hard Lefschetz Theorem (HLT) and the Hodge–Riemann bilinear relations (HRR) hold in various contexts: they impose restrictions on the cohomology algebra of a smooth compact Kähler manifold; they restrict the local monodromy of a polarized variation of Hodge structure; they impose conditions on the f-vectors of convex polytopes. While the statements of these theorems depend on the choice of a Kähler class, or its analog, there is usually a cone of possible choices. It is then natural to ask whether the HLT and HRR remain true in a mixed context. In this note, we present a unified approach to proving the mixed HLT and HRR, generalizing the known results, and proving it in new cases, such as the intersection cohomology of nonrational polytopes.