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Newton polyhedra and an algorithm for computing Hodge-Deligne numbers. (English. Russian original) Zbl 0669.14012
Math. USSR, Izv. 29, 279-298 (1987); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 50, No. 5, 925-945 (1986).
This paper is devoted to the description of invariants of mixed Hodge structure on algebraic varieties X in terms of Newton polyhedra. Let \(H^*_ c(X)\) be a compact cohomology of X, \(h^{p,q}(H^*_ c(X))\) be the Hodge-Deligne numbers of mixed Hodge structure on X. Define numbers \(e^{p,q}(X)\) and \(e^ p(X)\) by the \(formulas:\)
e\({}^{p,q}(X)=\sum_{k}(-1)^ kh^{p,q}(H_ c^ k)(X)) \) and \(e^ p(X)=\sum_{q}e^{p,q}(X).\)
The number \(e^{p,q}(X)\) is an additive invariant of X. - Results of the paper:
(1) Calculation of \(e^{p,q}(X)\) in terms of Newton polyhedra, when X is a nondegenerate complete intersection in a toroidal manifold.
(2) Algorithm for the calculation of \(e^{p,q}(X).\)
(3) Calculation of \(h^{p,q}\), when X is a nondegenerate complete intersection in a compact toroidal manifold.
(4) Algorithm for the calculation of \(h^{p,q}\) for a noncompact complete intersection X in the following cases:
X is a complete intersection in \(({\mathbb{C}}\setminus 0)^ n\) and the Newton polyhedra of all equations have maximal dimension.
X is a complete intersection in \({\mathbb{C}}^ n\), and the Newton polyhedra of all equations contains the origin and intersect all coordinate axes.
Reviewer: S.V.Chmutov

MSC:
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14M10 Complete intersections
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
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