The Birational Invariance of Fundamental Group Schemes

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 $X \rightarrow Y$.