Karl-Heinz Fieseler

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.

Semilinear elliptic problems and concentration compactness on non-compact Riemannian manifolds

Existence of solution for semilinear problem with the Laplace-Beltrami operator on non-compact Riemannian manifolds with rich symmetries is proved by concentration compactness based on actions of the manifolds isometry group.

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.

Towards a combinatorial intersection cohomology for fans

In this Note, the real intersection cohomology IH•(XΔ) of a toric variety XΔ is described in a purely combinatorial way using methods of elementary commutative algebra only. We define, for arbitrary fans, the notion of a “minimal extension sheaf” E• on the fan Δ as an axiomatic characterization of the equivariant intersection cohomology sheaf. This provides a purely algebraic interpretation of the h- and g-vector of an arbitrary polytope or fan under a natural vanishing condition. The results presented in this Note originate from joint work with G. Barthel, J.-P. Brasselet and L. Kaup (see [3]).

Hyperbolic ℂ*-Actions on Affine Algebraic Surfaces

We consider hyperbolic ℂ*-actions τ on normal affine algebraic surfaces W. It is well known how to describe the fixed point set F of τ using the topological Euler — Poincare characteristic of W. Our particular interest is to calculate more subtle geometric invariants of τ from homological data, like the multiplicities of the exceptional orbits or the patching weights of the fixed points. To that end we need the torsion groups of the (intersection) homology of W. A basic prerequisite is a description of the geometry of such ℂ*-surfaces in the analytic category, which we give in form of a “graph”. It includes the algebraic quotient Y (τ), the orbit data of the exceptional orbits and the patching weights of the fixed points, parametrized by a finite set B ⊂ Y of exceptional points. A parametrization of such surfaces in the algebraic setting is provided by H 1(Y (τ),O* alg).

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.

Caractérisation des variétés homologiques à l'aide des invariants d'homologie d'intersection

H. C. King [Ki], a donne un contre-exemple a cette conjecture et a montre qu’elle devient vraie si l’on permet des perversites plus generales que celles de M. Goresky et R. MacPherson, appelees “loose perversities”. Une autre approche consiste a montrer que la conjecture de M. Goresky et R. MacPherson est vraie pour les seules perversites portant leur nom mais en ajoutant une hypothese supplementaire. C’est par exemple le cas des pseudovarietes normales telles

Intersection Homology of C*-Surfaces

Publisher Summary This chapter discusses the intersection homology of ℂ*-surfaces. The chapter presents the comparison theorems between intersection homology and ordinary cohomology. It generalizes some known results using more systematically the singular Poincare duality homomorphism as constructed to get more information about the intersection homology groups. As a consequence of these results, the intersection homology I p H * ( Y ,ℤ) turns out to be a homotopy invariant in the class of compact pl -pseudomanifolds Y with isolated singularities. The chapter discusses the case of normal compact complex surfaces X . A nonsingular model of X is used to calculate the intersection torsion, and to compare it with the torsion in ordinary homology. The chapter also proves a formula for the determinant of the intersection pairing and gives conditions under which the covariant universal coefficient formula remains valid in intersection homology.