Andrzej Weber

Thom Polynomials of Invariant Cones, Schur Functions and Positivity

We generalize the notion of Thom polynomials from singularities of maps between two complex manifolds to invariant cones in representations, and collections of vector bundles. We prove that the generalized Thom polynomials, expanded in the products of Schur functions of the bundles, have nonnegative coefficients. For classical Thom polynomials associated with maps of complex manifolds, this gives an extension of our former result for stable singularities to nonnecessary stable ones. We also discuss some related aspects of Thom polynomials, which makes the article expository to some extent.

A morphism of intersection homology induced by an algebraic map

Let f : X → Y be a map of algebraic varieties. Barthel, Brasselet, Fieseler, Gabber and Kaup have shown that there exists a homomorphism of intersection homology groups f∗ : IH∗(Y ) → IH∗(X) compatible with the induced homomorphism on cohomology. The crucial point in the argument is reduction to the finite characteristic. We give an alternative and short proof of the existence of a homomorphism f∗. Our construction is an easy application of the Decomposition Theorem. Let X be an algebraic variety, IH∗(X) = H∗(X ; ICX) its rational intersection homology group with respect to the middle perversity and ICX the intersection homology sheaf which is an object of derived category of sheaves over X [GM1]. We have the homomorphism ωX : H∗(X ; Q) −−→ IH∗(X) induced by the canonical morphism of the sheaves ωX : QX −−→ ICX . Let f : X −→ Y be a map of algebraic varieties. It induces a homomorphism of the cohomology groups. The natural question arises: Does there exist an induced homomorphism for intersection homology compatible with f∗ ? IH∗(Y ) ? −−→ IH∗(X) xωY xωX H∗(Y ; Q) f ∗ −−→ H∗(X ; Q) . The answer is positive. For topological reasons the map in question exists for normally nonsingular maps [GM1, §5.4.3] and for placid maps [GM3, §4]. The authors of [BBFGK] proved the following: Theorem 1. Let f : X −→ Y be an algebraic map of algebraic varieties. Then there exists a morphism λf : ICY −→ Rf∗ICX such that the following diagram with the canonical morphisms commutes: ICY λf −−→ Rf∗ICX xωY xRf∗(ωX) QY αf −−→ Rf∗QX . Received by the editors February 24, 1998. 1991 Mathematics Subject Classification. Primary 14F32, 32S60; Secondary 14B05, 14C25.

Weights in the cohomology of toric varieties

We describe the weight filtration in the cohomology of toric varieties. We present a role of the Frobenius automorphism in an elementary way. We prove that equivariant intersection homology of an arbitrary toric variety is pure. We obtain results concerning Koszul duality: nonequivariant intersection cohomology is equal to the cohomology of the Koszul complexIHT*(X)⊗H*(T). We also describe the weight filtration inIH*(X).

Koszul duality for modules over Lie algebras

Let G be a compact Lie group. Set Λ• = H∗(G) and S • = H(BG). The coefficients are in R or C. Suppose G acts on a reasonable space X. In the paper [GKM] Goresky, Kottwitz and MacPherson established a duality between the ordinary cohomology which is a module over Λ• and equivariant cohomology which is a module over S • . This duality is on the level of chains, not on the level of cohomology. Koszul duality says that there is an equivalence of derived categories of Λ•–modules and S • –modules. One can lift the structure of an S • –module on H G(X) and the structure of a Λ•–module on H(X) to the level of chains in such a way that the obtained complexes correspond to each other under Koszul duality. Equivariant coefficients in

Formality of equivariant intersection cohomology of algebraic varieties

We present a proof that the equivariant intersection cohomology of any complete algebraic variety acted by a connected algebraic group G is a free module over H*(BG).

Leray residue for singular varieties

1. Residue form. We begin with recalling the construction of the Leray residue form [Le]. Let K be a smooth hypersurface of a complex manifold M . Suppose that K is locally given by an equation f = 0 where f is holomorphic and df does not vanish along K. Let ω ∈ Ω(M \K) be a complex valued C∞-form with the first order pole on K, i.e. f ω extends to a smooth form on M . If ω is closed then it can be written (locally) in the form

EQUIVARIANT HIRZEBRUCH CLASSES AND MOLIEN SERIES OF QUOTIENT SINGULARITIES

We study properties of the Hirzebruch class of quotient singularities ℂn/G, where G is a finite matrix group. The main result states that the Hirzebruch class coincides with the Molien series of G under suitable substitution of variables. The Hirzebruch class of a crepant resolution can be described specializing the orbifold elliptic genus constructed by Borisov and Libgober. It is equal to the combination of Molien series of centralizers of elements of G. This is an incarnation of the McKay correspondence. The results are illustrated with several examples, in particular of 4-dimensional symplectic quotient singularities.

On the Pełczyński conjecture on Auerbach bases

We consider Auerbach bases in Banach spaces of dimension n>2. We show that there exists at least (n-1)n/2+1 such bases. This estimate follows from the calculation of the Lusternik-Schnirelmann category of the flag variety. A better estimate is obtained for generic Banach spaces by the Morse theory.

Leray Spectral Sequence for Complements of Certain Arrangements of Smooth Submanifolds

Let X be a complex algebraic manifold. Let U be the complement of a configuration of submanifolds. We study the Leray spectral sequence of the inclusion U ↪ X computing the cohomology of U. Under some condition posed on the intersections of submanifolds we show that the Leray spectral sequence degenerates on E3. This result generalizes well known properties of hyperplane arrangements. The main cause which rigidifies the spectral sequence is the weight filtration in cohomology.

ON TORSION IN HOMOLOGY OF SINGULAR TORIC VARIETIES

Let