The naturality in Kirwan's decomposition
type: Item
instance of: scholarly article
title: The naturality in Kirwan's decomposition
zbMATH Open document ID: 0914.57024
author: Ching-Li Chai
author: Amnon Neeman
published in: Manuscripta Mathematica
publication date: 1999-03-09T00:00:00Z
review text: Let \(X\) be a compact symplectic manifold acted on by a compact Lie group \(K\) with a moment map \(\mu:X\to\text{Lie} (K)^*\). In [\textit{F. C. Kirwan}, Cohomology of quotients in symplectic and algebraic geometry, Math. Notes 31, Princeton (1984; Zbl 0553.14020)] it is shown that \(X\) has a natural stratification such that the associated spectral sequence for cohomology with rational coefficients degenerates at the \(E_2\) term. The authors of this note observe that the same proof shows that there is an integer \(N\), depending on the action of \(K\) on \(X\), such that the spectral sequence of the stratification degenerates for any complex oriented cohomology theory in which \(N\) is inverted. They pose two interesting problems and discuss their consequences. The first is whether the same result is still true if the assumption of complex orientability is removed. The second applies to the algebraic case when \(X\) is a projective variety acted on linearly by a reductive group \(G\) (which over the complex numbers can be thought of as the complexification of \(K)\), and asks how the degeneration of the spectral sequence for \(l\)-adic cohomology relates to the action of Galois groups.
Mathematics Subject Classification ID: 57S15
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 55N20
Mathematics Subject Classification ID: 55T99
zbMATH DE Number: 1239588
zbMATH Keywords: symplectic manifold
zbMATH Keywords: Hamiltonian group action
zbMATH Keywords: stratification
zbMATH Keywords: spectral sequence for cohomology
reviewed by: Frances Kirwan
MaRDI profile type: Publication
full work available at URL: s002290050112
OpenAlex ID: W1999325656
DOI: 10.1007/S002290050112
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description: scientific article; zbMATH DE number 1239588