Zariski Hyperplane Section Theorem for Grassmannian Varieties

http://dx.doi.org/10.4153/CJM-2003-007-9
Canad. J. Math. 55(2003), 157-180

  Published:2003-02-01
 Printed: Feb 2003

Let $\phi \colon X\to M$ be a morphism from a smooth irreducible complex quasi-projective 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 differential-geometric 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 $\pi_2 (\phi) \colon \pi_2 (X) \to \pi_2 (M)$.