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 Mosc. Math. J., 2002, Volume 2, Number 1, Pages 161–182 (Mi mmj50)

The Bott formula for toric varieties

E. N. Materov

Eberhard Karls Universität Tübingen

Abstract: The purpose of this paper is to give an explicit formula which allows one to compute the dimension of the cohomology groups of the sheaf $\Omega_{\mathbb P}^p(D)= \Omega_{\mathbb P}^p\otimes {\mathcal O_\mathbb P}(D)$ of $p$-th differential forms Zariski twisted by an ample invertible sheaf on a complete simplicial toric variety. The formula involves some combinatorial sums of integer points over all faces of the support polytope for ${\mathcal O_\mathbb P}(D)$. Comparison of two versions of the Bott formula gives some elegant corollaries in the combinatorics of simple polytopes. Also, we obtain a generalization of the reciprocity law. Some applications of the Bott formula are discussed.

Key words and phrases: $p$-th Hilbert–Ehrhart polynomial, Zariski forms.

DOI: https://doi.org/10.17323/1609-4514-2002-2-1-161-182

Full text: http://www.ams.org/.../abst2-1-2002.html
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MSC: Primary 14M25; Secondary 52B20, 52B11, 32L10, 58A10
Received: July 7, 2001; in revised form November 25, 2001
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Citation: E. N. Materov, “The Bott formula for toric varieties”, Mosc. Math. J., 2:1 (2002), 161–182

Citation in format AMSBIB
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This publication is cited in the following articles:
1. Rams S., “Defect and Hodge numbers of hypersurfaces”, Advances in Geometry, 8:2 (2008), 257–288
2. Mavlyutov A.R., “Cohomology of rational forms and a vanishing theorem on toric varieties”, Journal fur Die Reine und Angewandte Mathematik, 615 (2008), 45–58
3. Egorychev G.P., “Method of Coefficients: an algebraic characterization and recent applications”, Advances in Combinatorial Mathematics, 2009, 1–30
4. Allaud E., Fernandez J., “Non-Genericity of Infinitesimal Variations of Hodge Structures Arising in Some Geometric Contexts”, Proceedings of the Edinburgh Mathematical Society, 53:1 (2010), 13–29
5. Dickenstein A., Nill B., “A Simple Combinatorial Criterion for Projective Toric Manifolds with Dual Defect”, Math Res Lett, 17:3 (2010), 435–448
6. Balaji V., Barik P., Nagaraj D.S., “On Degenerations of Moduli of Hitchin Pairs”, Electron. Res. Announc. Math. Sci., 20 (2013), 103–108
7. Maxim L.G., Schuermann J., “Characteristic Classes of Singular Toric Varieties”, Commun. Pure Appl. Math., 68:12 (2015), 2177–2236
8. Di Natale C., Fatighenti E., Fiorenza D., “Hodge Theory and Deformations of Affine Cones of Subcanonical Projective Varieties”, J. Lond. Math. Soc.-Second Ser., 96:3 (2017), 524–544