Gottfried Barthel

Hodge-Riemann Relations for Polytopes: A Geometric Approach

The key to the Hard Lefschetz Theorem for combinatorial intersection cohomology of polytopes is to prove the Hodge-Riemann bilinear relations. In these notes, we strive to present an easily accessible proof. The strategy essentially follows the original approach of [Ka], applying induction a la [BreLu2], but our guiding principle here is to emphasize the geometry behind the algebraic arguments by consequently stressing polytopes rather than fans endowed with a strictly convex conewise linear function. It is our belief that this approach makes the exposition more transparent since polytopes are more appealing to our geometric intuition than convex functions on a fan.

Invariante Divisoren und Schnitthomologie von torischen Varietäten

In this article, we complete the interpretation of groups of classes of invariant divisors on a complex toric variety X of dimension n in terms of suitable (co-) homology groups. In [BBFK], we proved the following result (see Satz 1 below): Let Cl DivC(X) and Cl Div T W(X) denote the groups of classes of invariant Cartier resp. Weil divisors on X. If X is non degenerate (i.e., not equivariantly isomorphic to the product of a toric variety and a torus of positive dimension), then the natural homomorphisms Cl DivC(X) → H 2(X) and Cl DivW(X) → H cld 2n−2(X) are isomorphisms, the inclusion Cl DivC(X) ↪→ Cl Div T W(X) corresponds to the Poincare duality 1991 Mathematics Subject Classification: Primary 14M25, 14C20, 55N33; Secondary 52B20, 32M12. Unterstutzt durch das Programm ”PROCOPE“ zur Forderung der deutsch-franzosischen Kooperation in der Forschung. Avec le concours du programme «PROCOPE» pour l’avancement de la cooperation francoallemande dans la recherche scientifique. The paper is in final form and no version of it will be published elsewhere.

INTRODUCTION TO BASIC TORIC GEOMETRY

The aim of these notes is to give a concise introduction to some fundamental notions of toric geometry, with applications to singularity theory in mind. Toric varieties and their singularities provide a lot of particularly interesting examples: Though belonging to a restricted class, they illustrate many central concepts for the general study of algebraic varieties and singularities. Quoting from the introduction of [Ful], one may say that “toric varieties have provided a remarkably fertile testing ground for general theories”. Whereas a singular variety may not be “globally” toric, singularities often are “toroidal”, i.e., locally analytically equivalent to toric ones, so toric geometry can help for a better understanding even of non-toric singular varieties. In addition to that, for studying certain classes of non-toroidal singularities, methods of toric geometry turn out to be most useful, e.g., for the resolution of “non-degenerate complete intersection singularities”. As a key feature, toric varieties admit a surprisingly simple, yet elegant and powerful description that prominently uses objects from elementary convex and combinatorial geometry. These objects are “rational” convex polyhedral cones and compatible collections thereof, called “fans”, in a real vector space of dimension equal to the complex dimension of the variety.