11

Suppose that $X$ and $Y$ are smooth quasi-projective varieties over $\mathbb{C}$ with a holomorphic map $f:X \to Y$ inducing isomorphisms $f_* : H_i(X;\mathbb{Q}) \to H_i(Y;\mathbb{Q})$ for all $i \geq 0$. Suppose further that there is an action of a reductive complex Lie group $G$ on $X$ and $Y$, that $f$ is $G$-equivariant, and that orbits are closed in both cases.

Is it true that the induced map $X/G \to Y/G$ on quotient varieties also induces an isomorphism on $H_i(-;\mathbb{Q})$, possibly by imposing additional conditions?

jacob
  • 193
  • 7
  • I have heard the claim that a version of this statement is true using GIT quotients (dropping the requirement that orbits are closed) and integral homology. If this is indeed true, I’d expect a version with rational coefficients holds... but I also would also appreciate a source! – jacob Jun 15 '18 at 19:47
  • 1
    I like this question a lot (+1) perhaps adding the tag "equivariant maps" would help. – Andres Mejia Jun 15 '18 at 19:49
  • 1
    I can prove this when $G$ is finite (the rational homology of the quotient is identified with the $G$-equivariant homology, and there is a spectral sequence calculating $H_^G(X)$ from $H_(G;H_*(X))$). It seems likely true in general, but this proof cannot generalize - the key point is that the rational homology of $BH$ for $H$ finite is concentrated in degree zero. –  Jun 17 '18 at 17:41

0 Answers0