1,569 research outputs found
Weight filtration on the cohomology of complex analytic spaces
We extend Deligne's weight filtration to the integer cohomology of complex
analytic spaces (endowed with an equivalence class of compactifications). In
general, the weight filtration that we obtain is not part of a mixed Hodge
structure. Our purely geometric proof is based on cubical descent for
resolution of singularities and Poincar\'e-Verdier duality. Using similar
techniques, we introduce the singularity filtration on the cohomology of
compactificable analytic spaces. This is a new and natural analytic invariant
which does not depend on the equivalence class of compactifications and is
related to the weight filtration.Comment: examples added + minor correction
A motivic version of the theorem of Fontaine and Wintenberger
We prove the equivalence between the categories of motives of rigid analytic
varieties over a perfectoid field of mixed characteristic and over the
associated (tilted) perfectoid field of equal characteristic. This
can be considered as a motivic generalization of a theorem of Fontaine and
Wintenberger, claiming that the Galois groups of and are
isomorphic. A main tool for constructing the equivalence is Scholze's theory of
perfectoid spaces.Comment: Stable version added. Accepted for publication. 46 page
Describing toric varieties and their equivariant cohomology
Topologically, compact toric varieties can be constructed as identification
spaces: they are quotients of the product of a compact torus and the order
complex of the fan. We give a detailed proof of this fact, extend it to the
non-compact case and draw several, mostly cohomological conclusions.
In particular, we show that the equivariant integral cohomology of a toric
variety can be described in terms of piecewise polynomials on the fan if the
ordinary integral cohomology is concentrated in even degrees. This generalizes
a result of Bahri-Franz-Ray to the non-compact case. We also investigate
torsion phenomena in integral cohomology.Comment: 13 pages; minor change
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