Mixed Lefschetz Theorems and Hodge-Riemann Bilinear Relations

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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 Kahler 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 Kahler 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.