Hodge-Riemann polynomials
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866917063575994368 |
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| author | Lu, Qing Zheng, Weizhe |
| author_facet | Lu, Qing Zheng, Weizhe |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_16992 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hodge-Riemann polynomials Lu, Qing Zheng, Weizhe Algebraic Geometry Combinatorics 14C17 (Primary) 05A20, 05E14, 14J60 (Secondary) 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. |
| title | Hodge-Riemann polynomials |
| topic | Algebraic Geometry Combinatorics 14C17 (Primary) 05A20, 05E14, 14J60 (Secondary) |
| url | https://arxiv.org/abs/2506.16992 |