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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2403.17747 |
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| _version_ | 1866910436553654272 |
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| author | Maxim, Laurentiu Schürmann, Jörg |
| author_facet | Maxim, Laurentiu Schürmann, Jörg |
| contents | We give a cohomological and geometrical interpretation for the weighted Ehrhart theory of a full-dimensional lattice polytope $P$, with Laurent polynomial weights of geometric origin. For this purpose, we calculate the motivic Chern and Hirzebruch characteristic classes of a mixed Hodge module complex $\mathcal{M}$ whose underlying cohomology sheaves are constant on the $\mathbb{T}$-orbits of the toric variety $X_P$ associated to $P$. Besides motivic coefficients, this also applies to the intersection cohomology Hodge module. We introduce a corresponding generalized Hodge $χ_y$-polynomial of the ample divisor $D_P$ on $X_P$. Motivic properties of these characteristic classes are used to express this Hodge polynomial in terms of a very general weighed lattice point counting and the corresponding weighted Ehrhart theory. We introduce, for such a mixed Hodge modules complex $\mathcal{M}$ on $X$, an Ehrhart polynomial $E_{P,\mathcal{M}}$ generalizing the Hodge polynomial of $\mathcal{M}$ and satisfying a reciprocity formula and a purity formula fitting with the duality for mixed Hodge modules. This Ehrhart polynomial and its properties depend only on a Laurent polynomial weight function on the faces $Q$ of $P$. In the special case of the intersection cohomology mixed Hodge module, the weight function corresponds to Stanley's $g$-function of the polar polytope of $P$, hence it depends only on the combinatorics of $P$. In particular, we obtain a combinatorial formula for the intersection cohomology signature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_17747 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weighted Ehrhart theory via mixed Hodge modules on toric varieties Maxim, Laurentiu Schürmann, Jörg Algebraic Geometry We give a cohomological and geometrical interpretation for the weighted Ehrhart theory of a full-dimensional lattice polytope $P$, with Laurent polynomial weights of geometric origin. For this purpose, we calculate the motivic Chern and Hirzebruch characteristic classes of a mixed Hodge module complex $\mathcal{M}$ whose underlying cohomology sheaves are constant on the $\mathbb{T}$-orbits of the toric variety $X_P$ associated to $P$. Besides motivic coefficients, this also applies to the intersection cohomology Hodge module. We introduce a corresponding generalized Hodge $χ_y$-polynomial of the ample divisor $D_P$ on $X_P$. Motivic properties of these characteristic classes are used to express this Hodge polynomial in terms of a very general weighed lattice point counting and the corresponding weighted Ehrhart theory. We introduce, for such a mixed Hodge modules complex $\mathcal{M}$ on $X$, an Ehrhart polynomial $E_{P,\mathcal{M}}$ generalizing the Hodge polynomial of $\mathcal{M}$ and satisfying a reciprocity formula and a purity formula fitting with the duality for mixed Hodge modules. This Ehrhart polynomial and its properties depend only on a Laurent polynomial weight function on the faces $Q$ of $P$. In the special case of the intersection cohomology mixed Hodge module, the weight function corresponds to Stanley's $g$-function of the polar polytope of $P$, hence it depends only on the combinatorics of $P$. In particular, we obtain a combinatorial formula for the intersection cohomology signature. |
| title | Weighted Ehrhart theory via mixed Hodge modules on toric varieties |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2403.17747 |