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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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