Hodge-Riemann polynomials

Qing Lu; Weizhe Zheng

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arXiv · 2506.16992

Hodge-Riemann polynomials

Qing Lu·Weizhe Zheng

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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 any product of Schur polynomials is Schur log-concave, confirming conjectures of Ross and Wu.

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Qing Lu, Weizhe Zheng. 2025-06-20. Hodge-Riemann polynomials. https://arxiv.org/abs/2506.16992

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