Fundamental group of a geometric invariant theoretic quotient

TL;DR

This paper proves that the algebraic and topological fundamental groups of a smooth projective variety are preserved under geometric invariant theory quotients by reductive groups, establishing an isomorphism.

Contribution

It establishes the isomorphism of fundamental groups between a variety and its GIT quotient, extending known results to a broader class of algebraic varieties.

Findings

Algebraic fundamental groups are isomorphic under GIT quotients.

Topological fundamental groups are preserved over complex numbers.

The rational map induces an isomorphism between fundamental groups.

Abstract

Let $M$ be an irreducible smooth projective variety, defined over an algebraically closed field, equipped with an action of a connected reductive affine algebraic group $G$, and let ${\mathcal L}$ be a $G$--equivariant very ample line bundle on $M$. Assume that the GIT quotient $M/\!\!/G$ is a nonempty set. We prove that the homomorphism of algebraic fundamental groups $\pi_1(M)\, \longrightarrow\, \pi_1(M/\!\!/G)$, induced by the rational map $M\, \longrightarrow\, M/\!\!/G$, is an isomorphism. If $k\,=\, \mathbb C$, then we show that the above rational map $M\, \longrightarrow \, M/\!\!/G$ induces an isomorphism between the topological fundamental groups.