Article contents
Zariski Hyperplane Section Theorem for Grassmannian Varieties
Published online by Cambridge University Press: 20 November 2018
Abstract
Let $\phi :\,X\,\to \,M$ be a morphism from a smooth irreducible complex quasi-projective variety $X$ to a Grassmannian variety $M$ such that the image is of dimension ≥ 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 ${{\text{ }\!\!\pi\!\!\text{ }}_{1}}\left( {{\left( \gamma \,o\,\phi \right)}^{-1}}\,\left( M\,\backslash \,D \right) \right)$ is isomorphic to a central extension of ${{\pi }_{1}}\left( M\,\backslash \,D \right)\,\,\times \,{{\pi }_{1}}\left( X \right)$ by the cokernel of ${{\pi }_{2}}\left( \phi \right)\,:\,{{\pi }_{2}}\left( X \right)\,\to {{\pi }_{2}}\left( M \right)$.
- Type
- Research Article
- Information
- Copyright
- Copyright © Canadian Mathematical Society 2003
References
- 1
- Cited by