Mixed Lefschetz Theorems and Hodge-Riemann Bilinear Relations

2007/07/10 by Eduardo Cattani, Cattani, Eduardo · 4 citations
Mathematics · #14F43 #32G20 (Primary) #32Q15 #52B20 (Secondary) #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.0707.1352

openalex publication_date 2007/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Statements analogous to the Hard Lefschetz Theorem (HLT) and the Hodge-Riemann bilinear relations (HRR) hold in a variety of contexts: they impose restrictions on the cohomology algebra of a smooth compact Kähler manifold or on the intersection cohomology of a projective toric variety; they restrict the local monodromy of a polarized variation of Hodge structure; they impose conditions on the possible 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 Kähler classes. 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 previously known results, and proving it in new cases such as the intersection cohomology of non-rational polytopes.

Cited by

• A decomposition theorem for Lefschetz modules
• A Hessian criterion for totally nonnegative Toeplitz matrices and a theorem of Cattani
• Hodge-Riemann polynomials
• Wall crossing structure from quantum phenomena to Feynman Integrals

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