arXiv subjects

Matthias Franz

Publications and source records attributed to Matthias Franz.

40 records · Page 3Linked to original sources

arXiv2004

Weights in cohomology and the Eilenberg-Moore spectral sequence

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.

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arXiv2003

Koszul duality and equivariant cohomology

Let G be a topological group such that its homology H(G) with coefficients in a principal ideal domain R is an exterior algebra, generated in odd degrees. We show that the singular cochain functor carries the duality between G-spaces and spaces over BG to the Koszul duality between modules up to homotopy over H(G) and H^*(BG). This gives in particular a Cartan-type model for the equivariant cohomology of a G-space. As another corollary, we obtain a multiplicative quasi-isomorphism C^*(BG) -> H^*(BG). A key step in the proof is to show that a differential Hopf algebra is formal in the category of A-infinity algebras provided that it is free over R and its homology an exterior algebra.

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arXiv2003

On the integral cohomology of smooth toric varieties

Let $X_Σ$ be a smooth, not necessarily compact toric variety. We show that a certain complex, defined in terms of the fan $Σ$, computes the integral cohomology of $X_Σ$, including the module structure over the homology of the torus. In some cases we can also give the product. As a corollary we obtain that the cycle map from Chow groups to integral Borel-Moore homology is split injective for smooth toric varieties. Another result is that the differential algebra of singular cochains on the Borel construction of $X_Σ$ is formal.

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arXiv2003

Koszul duality and equivariant cohomology for tori

Let T be a torus. We show that Koszul duality can be used to compute the equivariant cohomology of topological T-spaces as well as the cohomology of pull backs of the universal T-bundle. The new features are that no further assumptions about the spaces are made and that the coefficient ring may be arbitrary. This gives in particular a Cartan-type model for the equivariant cohomology of a T-space with arbitrary coefficients. Our method works for intersection homology as well.

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