Hard Lefschetz theorem and Hodge-Riemann relations for intersection cohomology of nonrational polytopes

Details Hard Lefschetz theorem and Hodge-Riemann relations for intersection cohomology of nonrational polytopes Hard Lefschetz theorem and Hodge-Riemann relations for intersection cohomology of nonrational polytopes

2003-02-19

arxiv.org/abs/math/0302236v3

Mathematics

Algebraic Geometry

Final version to appear in Indiana University Mathematics Journal

Scientific paper

The Hard Lefschetz theorem for intersection cohomology of nonrational polytopes was recently proved by K. Karu [Ka]. This theorem implies the conjecture of R. Stanley on the unimodularity of the generalized $h$-vector. In this paper we strengthen Karu's theorem by introducing a canonical bilinear form $(\cdot ,\cdot)_{\Phi}$ on the intersection cohomology $IH(\Phi)$ of a complete fan $\Phi$ and proving the Hodge-Riemann bilinear relations for $(\cdot ,\cdot)_{\Phi}$.

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