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.