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Goresky–Pardon lifts of Chern classes and associated Tate extensions

Part of: Arithmetic problems. Diophantine geometry (Co)homology theory Singularities

Published online by Cambridge University Press:  03 May 2017

Eduard Looijenga
Eduard Looijenga*
Affiliation:
Yau Mathematical Sciences Center, Tsinghua University Beijing, China Mathematisch Instituut, Universiteit Utrecht, Nederland email [email protected]
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Abstract

Let $X$ be an irreducible complex-analytic variety, ${\mathcal{S}}$ a stratification of $X$ and ${\mathcal{F}}$ a holomorphic vector bundle on the open stratum ${X\unicode[STIX]{x0030A}}$. We give geometric conditions on ${\mathcal{S}}$ and ${\mathcal{F}}$ that produce a natural lift of the Chern class $\operatorname{c}_{k}({\mathcal{F}})\in H^{2k}({X\unicode[STIX]{x0030A}};\mathbb{C})$ to $H^{2k}(X;\mathbb{C})$, which, in the algebraic setting, is of Hodge level ${\geqslant}k$. When applied to the Baily–Borel compactification $X$ of a locally symmetric variety ${X\unicode[STIX]{x0030A}}$ and an automorphic vector bundle ${\mathcal{F}}$ on ${X\unicode[STIX]{x0030A}}$, this refines a theorem of Goresky–Pardon. In passing we define a class of simplicial resolutions of the Baily–Borel compactification that can be used to define its mixed Hodge structure. We use this to show that the stable cohomology of the Satake ($=$ Baily–Borel) compactification of ${\mathcal{A}}_{g}$ contains nontrivial Tate extensions.

Keywords

Baily–Borel compactificationGoresky–Pardon Chern classTate extension

MSC classification

Primary: 14G35: Modular and Shimura varieties 14F43: Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) 32S35: Mixed Hodge theory of singular varieties
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 7 , July 2017 , pp. 1349 - 1371
Copyright
© The Author 2017 

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