Danilov, V. I.; Khovanskij, A. G. 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 Cited in 4 ReviewsCited in 56 Documents 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) Keywords:toric variety; Hodge-Deligne numbers; mixed Hodge structure; Newton polyhedra; complete intersection; toroidal manifold PDFBibTeX XMLCite \textit{V. I. Danilov} and \textit{A. G. Khovanskij}, Math. USSR, Izv. 29, 279--298 (1987; Zbl 0669.14012); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 50, No. 5, 925--945 (1986) Full Text: DOI