On the Picard group of a compact flat projective variety

In this note, we describe the Picard group of the class of compact, smooth, flat, projective varieties. In view of Charlap's work and Johnson's characterization, we construct line bundles over such manifolds as the holonomy-invariant elements of the Neron-Severi group of a projective flat torus covering the manifold. We prove a generalized version of the Appell-Humbert theorem which shows that the nontrivial elements of the Picard group are precisely those coming from the above construction. Our calculations finally give an estimate for the set of positive line bundles for such varieties.

     

G-structure on the cohomology of Hopf algebras

We prove that is a Gerstenhaber algebra, where is a Hopf algebra. In case is the Drinfeld double of a finite-dimensional Hopf algebra , our results imply the existence of a Gerstenhaber bracket on . This fact was conjectured by R. Taillefer. The method consists of identifying as a Gerstenhaber subalgebra of (the Hochschild cohomology of ).

     

Thomason's theorem for varieties over algebraically closed fields

We present a novel proof of Thomason's theorem relating Bott inverted algebraic -theory with finite coefficients and étale cohomology for smooth varieties over algebraically closed ground fields. Our proof involves first introducing a new theory, which we term algebraic -homology, and proving it satisfies étale descent (with finite coefficients) on the category of normal, Cohen-Macaulay varieties. Then, we prove algebraic -homology and algebraic -theory (each taken with finite coefficients) coincide on smooth varieties upon inverting the Bott element.

     

Ideals of the cohomology rings of Hilbert schemes and their applications

We study the ideals of the rational cohomology ring of the Hilbert scheme of points on a smooth projective surface . As an application, for a large class of smooth quasi-projective surfaces , we show that every cup product structure constant of is independent of ; moreover, we obtain two sets of ring generators for the cohomology ring .

Similar results are established for the Chen-Ruan orbifold cohomology ring of the symmetric product. In particular, we prove a ring isomorphism between and for a large class of smooth quasi-projective surfaces with numerically trivial canonical class.

     

Cohomology operations for Lie algebras

If is a Lie algebra over and its centre, the natural inclusion extends to a representation of the exterior algebra of in the cohomology of . We begin a study of this representation by examining its Poincaré duality properties, its associated higher cohomology operations and its relevance to the toral rank conjecture. In particular, by using harmonic forms we show that the higher operations presented by Goresky, Kottwitz and MacPherson (1998) form a subalgebra of , and that they can be assembled to yield an explicit Hirsch-Brown model for the Borel construction associated to .

     

A Vanishing Theorem for the Tangential de Rham Cohomology of a Foliation with Amenable Fundamental Groupoid

For a space X carrying a foliation , the authors study the cohomology of the complex of differential forms which are smooth along the leaves and transversally locally L . It is shown that, if the leaves are nonpositively curved manifolds of rank at least r in a suitable uniform sense and the fundamental groupoid of the foliation is amenable, then the cohomology vanishes in degrees above r. This result is inspired by some of Gromov's results on bounded cohomology.     

A New Cohomology Theory Associated to Deformations of Lie Algebra Morphisms

We introduce a new cohomology theory related to deformations of Lie algebra morphisms. This notion involves simultaneous deformations of two Lie algebras and a homomorphism between them.This revised version was published online in March 2005 with corrections to the cover date.     

On D. Peterson's comparison formula for Gromov-Witten invariants of

We prove a formula of Dale Peterson comparing Gromov-Witten (GW) invariants of to those of using canonical reductions of bundles.

     

K-theory tools for local and asymptotic cyclic cohomology

A generalization of the Connes-Thom isomorphism is given for stable, homotopy invariant, and split exact functors on separable -algebras. As examples of these functors, we concentrate on asymptotic and local cyclic cohomology, and the result is applied to improve some formulas in asymptotic and local cyclic cohomology of -algebras. As another application, it is shown that these cyclic theories are rigid under Rieffel's deformation quantizations.

     

Local Cohomology of BP*BP-Comodules

Given a spectrum X, we construct a spectral sequence of BP_*BP-comodulesthat converges to BP_*(L_nX), where L_nX is the Bousfield localizationof X with respect to the Johnson–Wilson theory E(n)_*.The E₂-term of this spectral sequence consists of the derivedfunctors of an algebraic version of L_n. We show how to calculatethese derived functors, which are closely related to local cohomologyof BP_*-modules with respect to the ideal I_n+1. 2000 MathematicsSubject Classification 55N22, 55P60, 16W30.