The Birational Invariance Of Fundamental Group Schemes

Abstract

Let $k$ be a field, $f:X\to Y$ 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 ${\mathcal{C}}_X\subset \mathfrak{Vect}(X)$ and ${\mathcal{C}}_Y\subset \mathfrak{Vect}(Y)$, denote by $\pi ({\mathcal{C}}_X,x)$ and $\pi ({\mathcal{C}}_Y,y)$ the corresponding Tannaka group schemes. We establish a unified Tannakian criteria for the natural homomorphism $\pi ({\mathcal{C}}_X,x)\to \pi ({\mathcal{C}}_Y,y)$ to be an isomorphism. As applications, for a birational map $X⇢Y$ between smooth projective varieties over a perfect field $k$, we prove that there exists a natural isomorphism ${\pi }^{\ast }(X,x)\cong {\pi }^{\ast }(Y,y)$ for any $\ast \in \{S,N,EN,F,EF,Loc,ELoc,\acute{e}t,E\acute{e}t,uni\}$. In particular, we prove that the induced homomorphism ${\pi }^{str}(X,x)\to {\pi }^{str}(Y,y)$ is an isomorphism for any birational morphism $X\to Y$.

8 pages, Comments welcome!