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 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 π₁ (M) → π₁ (M//G), induced by the rational map M→ M//G, is an isomorphism. If k = C, then we show that the above rational map M → M//G induces an isomorphism between the topological fundamental groups.