Fundamental group of a geometric invariant theoretic quotient

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 ${\pi }_1(M)⟶{\pi }_1(M//G)$, induced by the rational map $M⟶M//G$, is an isomorphism. If $k=\mathbb{C}$, then we show that the above rational map $M⟶M//G$ induces an isomorphism between the topological fundamental groups.

12 pages, final version to appear in Transformation Groups