Hyperbolic localization of intersection cohomology

For a normal variety X defined over an algebraically closed field with an action of the multiplicative group T = G_m, we consider the "hyperbolic localization" functor D^b(X) → D^b(X^T), which localizes using closed supports in the directions flowing into the fixed points, and compact supports in the directions flowing out. We show that the hyperbolic localization of the intersection cohomology sheaf is a direct sum of intersection cohomology sheaves.     

The primitive cohomology lattice of a complete intersection

We describe the primitive cohomology lattice of a smooth even-dimensional complete intersection in projective space. To cite this article: A. Beauville, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Basic intersection cohomology of conical fibrations

We introduce notions of singular fibration and singular Seifert fibration. These notions naturally generalize that o locally trivial fibration to the category of stratified pseudomanifolds. For singular foliations determined by such fibrations, we prove the de Rham theorem for basic intersection cohomology recently introduced by the present authors. One of the main examples of such a structure is the natural projection to the space of fibers of a singular Riemannian foliation determined by a Lie group action on a compact smooth manifold.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 235–257.Original Russian Text Copyright © 2005 by M. Saralegi-Aranguren, R. Wolak.This revised version was published online in April 2005 with a corrected issue number.     

From moment graphs to intersection cohomology

We describe a method of computing equivariant and ordinary intersection cohomology of certain varieties with actions of algebraic tori, in terms of structure of the zero- and one-dimensional orbits. The class of varieties to which our formula applies includes Schubert varieties in flag varieties and affine flag varieties. We also prove a monotonicity result on local intersection cohomology stalks. Received: 9 November 2000 / Published online: 24 September 2001     

Basic intersection cohomology of conical fibrations

We introduce notions of singular fibration and singular Seifert fibration. These notions naturally generalize that o locally trivial fibration to the category of stratified pseudomanifolds. For singular foliations determined by such fibrations, we prove the de Rham theorem for basic intersection cohomology recently introduced by the present authors. One of the main examples of such a structure is the natural projection to the space of fibers of a singular Riemannian foliation determined by a Lie group action on a compact smooth manifold.     

The cohomology ring of a monomial algebra

Summary In this paper we study the algebra structure of the cohomology ring of a monomial algebra. This article was processed by the author using the IAT_EX style filecljour1 from Springer-Verlag.     

Vanishing of odd dimensional intersection cohomology II

For a variety where a connected linear algebraic group acts with only finitely many orbits, each of which admits an attractive slice, we show that the stratification by orbits is perfect for equivariant intersection cohomology with respect to any equivariant local system. This applies to provide a relationship between the vanishing of the odd dimensional intersection cohomology sheaves and of the odd dimensional global intersection cohomology groups. For example, we show that odd dimensional intersection cohomology sheaves and global intersection cohomology groups vanish for all complex spherical varieties. Received: 25 February 2000 / Accepted: 15 February 2001 / Published online: 23 July 2001     

The intersection cohomology of Schubert varieties is a combinatorial invariant

We give an explicit and entirely poset-theoretic way to compute, for any permutation v, all the Kazhdan–Lusztig polynomials P_x,y for x,y≤v, starting from the Bruhat interval [e,v] as an abstract poset. This proves, in particular, that the intersection cohomology of Schubert varieties depends only on the inclusion relations between the closures of its Schubert cells.     

Modular intersection cohomology complexes on flag varieties

We present a combinatorial procedure (based on the W-graph of the Coxeter group) which shows that the characters of many intersection cohomology complexes on low rank complex flag varieties with coefficients in an arbitrary field are given by Kazhdan–Lusztig basis elements. Our procedure exploits the existence and uniqueness of parity sheaves. In particular we are able to show that the characters of all intersection cohomology complexes with coefficients in a field on the flag variety of type A _n for n < 7 are given by Kazhdan–Lusztig basis elements. By results of Soergel, this implies a part of Lusztig’s conjecture for SL(n) with n ≤ 7. We also give examples where our techniques fail. In the appendix by Tom Braden examples are given of intersection cohomology complexes on the flag varities for SL(8) and SO(8) which have torsion in their stalks or costalks.     

The cohomology ring of the colored braid group

The cohomology ring is obtained for the space of ordered sets of n different points of a plane.Translated from Matematicheskie Zametki, Vol. 5, No. 2, pp. 227–231, February, 1969.The author thanks V. P. Palamodov and D. B. Fuks for useful discussions.     

The Hochschild cohomology ring of a cyclic block

Suppose is a block of a group algebra with cyclic defect group. We calculate the Hochschild cohomology ring of , giving a complete set of generators and relations. We then show that if is the principal block, the canonical map from to the Hochschild cohomology ring of induces an isomorphism modulo radicals.     

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 is a free module over .

     

Homology and cohomology intersection numbers of GKZ systems

We describe the homology intersection form associated to regular holonomic GKZ systems in terms of the combinatorics of regular triangulations. Combining this result with the twisted period relation, we obtain a formula of cohomology intersection numbers in terms of a Laurent series. We show that the cohomology intersection number depends rationally on the parameters. We also prove a conjecture of F. Beukers and C. Verschoor on the signature of the monodromy invariant hermitian form.     

Graded Witt ring and unramified cohomology

It is proved that for a smooth affine curveX over a local ring or global field, the graded Witt ring ofX is isomorphic to the graded unramified cohomology ring ofX. IfX is projective and has a rational point, the same result holds if and only if every quadratic space defined on the complement of a rational point extends toX. Such an extension is possible, for instance, if the canonical line bundle onX is a square in PicX.     

Superperverse intersection cohomology: stratification (in)dependence

Within its traditional range of perversity parameters, intersection cohomology is a topological invariant of pseudomanifolds. This is no longer true once one allows superperversities, perversities with . In this case, intersection cohomology may depend on the choice of the stratification by which it is defined. Topological invariance also does not hold if one allows stratifications with codimension one strata. Nonetheless, both errant situations arise in important situations, the former in the Cappell-Shaneson superduality theorem and the latter in any discussion of pseudomanifold bordism. We show that while full invariance of intersection cohomology under restratification does not hold in this generality, it does hold up to restratifications that fix the the top stratum.     

The ring structure on the cohomology of coordinate subspace arrangements

Every simplicial complex on the vertex set defines a real resp. complex arrangement of coordinate subspaces in resp. via the correspondence The linear structure of the cohomology of the complement of such an arrangement is explicitly given in terms of the combinatorics of and its links by the Goresky–MacPherson formula. Here we derive, by combinatorial means, the ring structure on the integral cohomology in terms of data of . We provide a non-trivial example of different cohomology rings in the real and complex case. Furthermore, we give an example of a coordinate arrangement that yields non-trivial multiplication of torsion elements. Received March 3, 1999; in final form June 24, 1999