## New answers tagged cohomology

9
votes

Accepted

### A question regarding isomorphism in cohomology for moduli space of stable bundles over a compact Riemann surface

Things are actually simpler. View $\Gamma _n=H^1(X,\mathbb{Z}/n)$ as the group of line bundles $L\in \operatorname{Pic}^{0}(X) $ with $L^{{\tiny \otimes }n}=\mathscr{O}_X$. The
map $N_0(n,k) \times \...

0
votes

### Models for computing cohomology of Lie groupoids

This is 2 years late, but since nobody has answered I'll give it a go.
$1.$ I don't know the answer to this.
$3.$ Yes, you can take cohomolgy of Lie groupoids with respect to sheaves, just like you ...

2
votes

Accepted

### Comparing cohomology of quotient by algebraic group and Borel subgroup

The spectral sequence degenerates on the second page since $X/B \to X/G$ is a smooth projective morphism (as $G/B$ is smooth projective) by a result of Deligne and Blanchard.
The local systems are ...

1
vote

### Holomorphic vector fields acting on Dolbeault cohomology

Klemyatin proved that this action is trivial if the corresponding
${\Bbb C}$-flow is compatible with some metric (hence can be extended
to a compact torus action),
https://arxiv.org/abs/1909.04075,
(N....

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