The naturality in Kirwan's decomposition
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.
scientific article; zbMATH DE number 1239588
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