[Paper Review] Monodromie d'une famille d'hypersurfaces

Ania Otwinowska|arXiv (Cornell University)|Mar 9, 2004

Mathematics and Applications3 citations

TL;DR

This paper extends classical monodromy results for families of smooth hypersurfaces in a complex projective manifold to the case where the hypersurfaces contain a fixed closed subscheme $W$ of dimension $n$. It establishes that for $N \geq 2n$ and sufficiently large degree $d \geq Ce$, the monodromy representation on the orthogonal complement of a natural subspace $H(W)$ in $H^N(X_F, \mathbb{C})$ is irreducible, generalizing the Lefschetz irreducibility theorem to singular subvarieties under degree bounds.

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.

Motivation & Objective

• To generalize classical monodromy results for smooth hypersurfaces to families containing a fixed closed subscheme $W$.
• To understand the monodromy action on the middle cohomology $H^N(X_F, \mathbb{C})$ of a smooth hypersurface $X_F$ containing $W$.
• To determine conditions under which the monodromy representation on the orthogonal complement of $H(W)$ is irreducible.
• To establish a degree threshold $d \geq Ce$ ensuring irreducibility, with $C$ depending only on the ambient variety $Y$.
• To explore the validity of the monodromy irreducibility result when $W$ is singular, extending beyond the smooth case.

Proposed method

• Define $\mathcal{V}^d(W)$ as the space of smooth hypersurfaces of degree $d$ containing a fixed closed subscheme $W$ of dimension $n$.
• Introduce $H(W) \subset H^N(X_F, \mathbb{C})$ as the subspace spanned by pullbacks from $Y$ and irreducible components of $W$ when $N=2n$, and decompose $H^N(X_F, \mathbb{C}) = H(W) \oplus H(W)^\perp$ via Poincaré duality and non-degeneracy of the intersection form.
• Use the hard Lefschetz theorem and the theory of degenerations to analyze monodromy on $H(W)^\perp$, particularly via the Picard-Lefschetz formula.
• Establish asymptotic estimates on Betti numbers and Euler characteristics using Chern classes and generating functions for the cohomology of hypersurfaces.
• Prove that for $d \geq Ce$, the monodromy representation on $H(W)^\perp$ is irreducible, with $C$ depending only on $Y$, by analyzing Hodge filtrations and the structure of the monodromy action.
• Handle the singular case by showing that the standard Lefschetz-type argument fails due to non-ordinary singularities, but conjecture that irreducibility still holds with $C=1$.

Experimental results

Research questions

• RQ1 Under what conditions is the space $\mathcal{V}^d(W)$ of smooth degree-$d$ hypersurfaces containing a fixed subscheme $W$ non-empty?
• RQ2 How does the monodromy action on $H^N(X_F, \mathbb{C})$ decompose when $X_F$ contains a fixed subscheme $W$?
• RQ3 What is the precise degree threshold $d \geq Ce$ ensuring that the monodromy representation on $H(W)^\perp$ is irreducible?
• RQ4 Does the monodromy irreducibility result extend to the case where $W$ is singular, particularly when $W$ has non-ordinary singularities?
• RQ5 Can the monodromy on $H(W)^\perp$ be shown to be irreducible with $C=1$ even when $W$ is singular, despite the failure of the standard Lefschetz degeneration argument?

Key findings

• If $N < 2n$, then $\mathcal{V}^d(W)$ is empty for all $d \geq e$, where $e$ is the degree bound for the ideal $I_W$.
• For $N \geq 2n$, the monodromy representation on $H(W)^\perp \subset H^N(X_F, \mathbb{C})$ is irreducible for all $d \geq Ce$, where $C \in \mathbb{R}^*_+$ depends only on $Y$.
• The subspace $H(W)$ is $\pi_1(\mathcal{V}^d(W), F)$-invariant and the decomposition $H^N(X_F, \mathbb{C}) = H(W) \oplus H(W)^\perp$ is orthogonal with respect to the intersection form.
• The irreducibility of monodromy on $H(W)^\perp$ is established via asymptotic estimates on Betti numbers and Euler characteristics, using Chern classes and generating functions.
• When $W$ is smooth, the result holds with $C=1$ due to the existence of a Lefschetz-type degeneration via blow-up, but this argument fails for singular $W$.
• The paper conjectures that the monodromy on $H(W)^\perp$ remains irreducible with $C=1$ even for singular $W$, though the standard proof technique does not extend.