Generalizing the notion of Newton polytope, we define the NewtonOkounkov body, respectively, for semigroups of integral points, graded algebras and linear series on varieties. We prove that any semigroup in the lattice Z n is asymptotically approximated by the semigroup of all the points in a sublattice and lying in a convex cone. Applying this we obtain several results. We show that for a large class of graded algebras, the Hilbert functions have polynomial growth and their growth coefficients satisfy a Brunn-Minkowski type inequality. We prove analogues of the Fujita approximation theorem for semigroups of integral points and graded algebras, which imply a generalization of this theorem for arbitrary linear series. Applications to intersection theory include a far-reaching generalization of the Kushnirenko theorem (from Newton polytope theory) and a new version of the Hodge inequality. We also give elementary proofs of the Alexandrov-Fenchel inequality in convex geometry and its analogue in algebraic geometry.

An algorithm is given for computing the mixed Hodge structure (more precisely, the Hodge-Deligne numbers) for cohomology of complete intersections in toric varieties in terms of Newton polyhedra specifying the complete intersection. In some particular cases the algorithm leads to explicit formulas. Bibliography: 8 titles. Homology and cohomology yield one of the most natural topological invariants of an arbitrary variety. It had been known since Hodge's work that the cohomology of projective complex manifolds admits a natural Hodge decomposition. Deligne (see [1] and [2]) generalized Hodge theory to arbitrary complex algebraic varieties by endowing their cohomology with a so-called mixed Hodge structure. In particular, for an arbitrary algebraic variety X and a Hodge structure H k (X) he defined the Hodge-Deligne numbers h p ' q (H k (X)). The present paper is devoted to a computation of these invariants for varieties X defined in (C\0) n by a system of polynomial equations f x = • • • = f k = 0, where the polynomials are nondegenerate with respect to their Newton polyhedra Δ ι ,...,Δ Ιί. We recall that the Newton polyhedron of a polynomial / = Ea m x m , where m = (m x ,...,m n) and χ = (x 1 ,..., x n), is the convex hull of those points m e R" for which a m Φ 0. The Newton polyhedron of a polynomial generalizes the notion of degree and plays a similar role. A system of equations with fixed Newton polyhedra is nondegenerate for almost all values of the coefficients. Throughout this paper the space (C\0)" is called the «-dimensional torus and is denoted by T". It might seem somewhat artificial that we study complete intersections in the torus T" and not in the affine space C". We suggest two reasons to justify our choice. First, cutting C " by coordinate hyperplanes into tori of various dimensions, we can apply the results on complete intersections in tori to complete intersections in C". This principle can be applied not only to the space C", but to arbitrary toric varieties, to wit algebraic varieties

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