Monodromie d'une famille d'hypersurfaces

TL;DR

This paper studies the monodromy of smooth hypersurfaces in a fixed variety, showing invariance of certain cohomology classes and irreducibility of the monodromy representation on their orthogonal complement, using topological methods.

Contribution

It characterizes the monodromy action on the cohomology of hypersurfaces containing a fixed subvariety, including cases with singularities, and proves irreducibility of a key monodromy representation.

Findings

Cohomology classes from the ambient variety and subvariety are monodromy invariant.

The monodromy representation on the orthogonal complement is irreducible.

The proof is topological, handling arbitrary singularities of the subvariety.

Abstract

I describe the monodromy of smooth hypersurfaces $X$ of high degree in a fixed smooth variety $Y$ containing a fixed subvariety $W$ of $Y$. The cohomology of $X$ in middle degree spanned by the pull-back of the cohomology of $Y$ and by the classes of the irreducible components of $W$ is monodromy invariant. I show that the monodromy representation on the orthogonal of those classes is irreducible. The proof is essentially topological. Difficulties arise from the fact that $W$ may have arbitrary singularities.