Newton Polyhedra and Good Compactification Theorem
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Newton Polyhedra and Good Compactification Theorem Askold Khovanskii1 Received: 14 February 2020 / Revised: 8 August 2020 / Accepted: 14 August 2020 © Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2020
Abstract A new transparent proof of the well-known good compactification theorem for the complex torus (C∗ )n is presented. This theorem provides a powerful tool in enumerative geometry for subvarieties in the complex torus. The paper also contains an algorithm constructing a good compactification for a subvariety in (C∗ )n explicitly defined by a system of equations. A new theorem on a toroidal-like compactification is stated. A transparent proof of this generalization of the good compactification theorem which is similar to proofs and constructions from this paper will be presented in a forthcoming publication. Keywords Good compactification theorem · Complex torus · Newton polyhedra · Toric variety
1 Introduction The paper is dedicated to a simple constructive proof of the good compactification theorem for the group (C∗ )n and to related elementary geometry of this group. A few words about the introduction. We briefly talk about the ring of conditions R(T n ) for the complex torus (C∗ )n in the introduction only. This ring suggests a version of intersection theory for algebraic cycles in (C∗ )n . We discuss a role of the good compactification theorem in this theory and explain how Newton polyhedra are related to the ring R(T n ). In Sect. 1.4, we state a new stronger version of the good compactification theorem. In the Sects. 1.5 and 1.6, we summarize the remainder of the paper and fix some notation.
Dedicated to Alexander Varchenko’s 70th birthday The work was partially supported by the NSERC Grant no. 156833-17.
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Askold Khovanskii [email protected] University of Toronto, Toronto, Canada
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1.1 The Ring of Conditions The good compactification theorem for a spherical homogeneous space U allows to define the ring of conditions of U (see [3] or the next page for the relevant definitions and statements). In the paper, we consider the case when U is a complex torus (C∗ )n equipped with the natural action of the torus on itself. Following the original ideas of Schubert, in the early 1980s, De Concini and Procesi developed an intersection theory for algebraic cycles in a symmetric homogeneous space [3]. Their theory, named the ring of conditions of U , can be automatically generalized to a spherical homogeneous space U . De Concini and Procesi showed that the description of such a ring can be reduced to homology rings (or to Chow rings) of an increasing chain of smooth G-equivariant compactifications of U . The ring of conditions R(U ) = R0 (U )+· · ·+Rn (U ) is a commutative graded ring with homogeneous components of degrees 0, . . . , n where n equals to the dimension of U . The component Rk (U ) consists of algebraic cycles Z k in U of codimension k, considered up to an equivalents relation ∼. An element in Z k is a formal sum of algebraic subvarieties of codim
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