The Birational Invariance of Fundamental Group Schemes

TL;DR

This paper proves that certain fundamental group schemes are invariant under birational transformations of smooth projective varieties over a perfect field, using Tannakian category criteria.

Contribution

It establishes general Tannakian conditions for the invariance of fundamental group schemes under birational morphisms.

Findings

Natural isomorphisms of various fundamental group schemes under birational maps.

The induced homomorphism between stratified fundamental groups is an isomorphism.

Invariance holds for a wide class of fundamental group schemes, including étale and unipotent types.

Abstract

Let $k$ be a field, $f \colon X \to Y$ a birational morphism of integral connected schemes proper over $k$ with $Y$ normal, $x \in X(k)$ lying over $y \in Y(k)$. For Tannakian categories $\cC_X \subset \Vect(X)$ and $\cC_Y \subset \Vect(Y)$, denote by $\pi(\cC_X,x)$ and $\pi(\cC_Y,y)$ the corresponding Tannaka group schemes. We establish general Tannakian criteria for the natural homomorphism $\pi(\cC_X,x)\to \pi(\cC_Y,y)$ to be an isomorphism. As applications, for a birational map $X \dashrightarrow Y$ between smooth projective varieties over a perfect field $k$, we prove that there exists a natural isomorphism $\pi^{*}(X,x)\cong \pi^{*}(Y,y)$ for any $* \in \{S,N,EN,F,EF,Loc,ELoc,\acute{e}t, E\acute{e}t,uni\}$. In particular, we prove that the induced homomorphism $\pi^{str}(X,x)\to \pi^{str}(Y,y)$ is an isomorphism for any birational morphism $ X \rightarrow Y$.