Weights in the cohomology of toric varieties

Weber, Andrzej

doi:10.2478/bf02475240

Cited by 4 publications

(1 citation statement)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This property deserves to be called "formality" (or maybe BG-formality), but unfortunately the word "formality" has been reserved for something else (namely freeness over H * (BG)). See the remarks in the introduction to [30].…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

Weights in cohomology and the Eilenberg-Moore spectral sequence

Franz¹,

Weber²

2005

Annales De L’institut Fourier

Self Cite

View full text Add to dashboard Cite

We show that in the category of complex algebraic varieties, the Eilenberg--Moore spectral sequence can be endowed with a weight filtration. This implies that it degenerates if all involved spaces have pure cohomology. As application, we compute the rational cohomology of an algebraic $G$-variety $X$ ($G$ being a connected algebraic group) in terms of its equivariant cohomology provided that $H_G(X)$ is pure. This is the case, for example, if $X$ is smooth and has only finitely many orbits. We work in the category of mixed sheaves; therefore our results apply equally to (equivariant) intersection homology

show abstract

Section: Annales De L'institut Fouriermentioning

confidence: 99%

Weights in cohomology and the Eilenberg-Moore spectral sequence

Franz¹,

Weber²

2005

Annales De L’institut Fourier

Self Cite

View full text Add to dashboard Cite

show abstract

Weights on cohomology, invariants of singularities, and dual complexes

2013

View full text Add to dashboard Cite

Abstract. We study the weight filtration on the cohomology of a proper complex algebraic variety and obtain natural upper bounds on its size, when it is the exceptional divisor of a singularity. We also give bounds for the cohomology of links. The invariants of singularities introduced here gives rather strong information about the topology of rational and related singularities.Given a divisor on a variety, the combinatorics governing the way the components intersect is encoded by the associated dual complex. This is the simplicial complex with p-simplices corresponding to (p + 1)-fold intersections of components of the divisor. Kontsevich and Soibelman [KS, A.4] and Stepanov [Stp1] had independently observed that the homotopy type of the dual complex of a simple normal crossing exceptional divisor associated to a resolution of an isolated singularity is an invariant for the singularity. In fact, [KS], and later Thuillier [T] and Payne [P] have obtained homotopy invariance results for more general dual complexes, such as those arising from boundary divisors. In characteristic zero, all these results are consequences of the weak factorization theorem of W lodarczyk [Wlo], and Abramovich-Karu-Matsuki-W lodarczyk [AKMW]; Thuillier uses rather different methods based on Berkovich's non-Archimedean analytic geometry. As we show here, a slight refinement of factorization (theorems 7.6, 7.7) and of these techniques yields some generalizations this statement. This applies to divisors of resolutions of arbitrary not necessarily isolated singularities, and even in a more general context (discussed in the final section). We also allow dual complexes associated with nondivisorial varieties, such as fibres of resolutions of nonisolated singularites. Here is a slightly imprecise formulation of theorems 7.5 and 7.9. [KS], although we were unaware of this paper at the time these results were completed).To put the remaining results in context, we note that the present paper can be considered as the extended version of our earlier preprint [ABW], where we gave

show abstract