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Hodge-Riemann polynomials (2506.16992v1)

Published 20 Jun 2025 in math.AG and math.CO

Abstract: We show that Schur classes of ample vector bundles on smooth projective varieties satisfy Hodge--Riemann relations on $H{p,q}$ under the assumption that $H{p-2,q-2}$ vanishes. More generally, we study Hodge--Riemann polynomials, which are partially symmetric polynomials that produce cohomology classes satisfying the Hodge--Riemann property when evaluated at Chern roots of ample vector bundles. In the case of line bundles and in bidegree $(1,1)$, these are precisely the nonzero dually Lorentzian polynomials. We prove various properties of Hodge--Riemann polynomials, confirming predictions and answering questions of Ross and Toma. As an application, we show that the derivative sequence of a product of Schur polynomials is Schur log-concave, confirming conjectures of Ross and Wu.

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