Abstract view
Zariski Hyperplane Section Theorem for Grassmannian Varieties


Published:20030201
Printed: Feb 2003
Abstract
Let $\phi \colon X\to M$ be a morphism from a smooth irreducible
complex quasiprojective variety $X$ to a Grassmannian variety $M$
such that the image is of dimension $\ge 2$. Let $D$ be a reduced
hypersurface in $M$, and $\gamma$ a general linear automorphism of
$M$. We show that, under a certain differentialgeometric condition
on $\phi(X)$ and $D$, the fundamental group $\pi_1 \bigl( (\gamma
\circ \phi)^{1} (M\setminus D) \bigr)$ is isomorphic to a central
extension of $\pi_1 (M\setminus D) \times \pi_1 (X)$ by the cokernel
of $\pi_2 (\phi) \colon \pi_2 (X) \to \pi_2 (M)$.