[Paper Review] The monodromy theorem for compact Kähler manifolds and smooth quasi-projective varieties

Nero Budur, Yongqiang Liu+1 more|arXiv (Cornell University)|Sep 21, 2016

Algebraic Geometry and Number Theory11 references4 citations

TL;DR

This paper establishes a monodromy-type theorem for compact Kähler manifolds and smooth quasi-projective varieties by analyzing the action of the infinite cyclic cover's deck transformation on homology with complex coefficients. It proves that for compact Kähler manifolds, all Jordan blocks of the monodromy action on the torsion part of homology are of size one, while for smooth quasi-projective varieties, an upper bound on Jordan block sizes is given, generalizing the classical local Milnor monodromy theorem.

ABSTRACT

Given any connected topological space $X$, assume that there exists an epimorphism $ϕ: π_1(X) o \mathbb{Z}$. The deck transformation group $\mathbb{Z}$ acts on the associated infinite cyclic cover $X^ϕ$ of $X$, hence on the homology group $H_i(X^ϕ, \mathbb{C})$. This action induces a linear automorphism on the torsion part of the homology group as a module over the Laurent ring $\mathbb{C}[t,t^{-1}]$, which is a finite dimensional $\mathbb{C}$-vector space. We study the sizes of the Jordan blocks of this linear automorphism. When $X$ is a compact Kähler manifold, we show that all the Jordan blocks are of size one. When $X$ is a smooth complex quasi-projective variety, we give an upper bound on the sizes of the Jordan blocks, which is an analogue of the Monodromy Theorem for the local Milnor fibration.

Motivation & Objective

• To extend the classical Monodromy Theorem from local Milnor fibrations to global settings on compact Kähler manifolds and smooth quasi-projective varieties.
• To analyze the monodromy action on the homology of infinite cyclic covers associated with epimorphisms π₁(X) → ℤ.
• To determine the maximal size of Jordan blocks in the monodromy action on the torsion part of homology with complex coefficients.
• To establish a bound on Jordan block sizes for smooth quasi-projective varieties, analogous to the local Milnor case.
• To use Hodge theory and spectral sequences to prove that the cohomology of the infinite cyclic cover admits a filtration with semi-simple graded quotients.

Proposed method

• Define the infinite cyclic cover X^φ associated with an epimorphism φ: π₁(X) → ℤ, and study the deck transformation action on H_i(X^φ, ℂ).
• Model the homology as a module over the Laurent polynomial ring R = ℂ[t, t⁻¹], focusing on the torsion submodule T_i(X, φ).
• Decompose T_i(X, φ) into generalized eigenspaces T_i^λ(X, φ) corresponding to eigenvalues λ ∈ ℂ*, and define the Jordan block size S_λ as the minimal m such that (t - λ)^m annihilates the module.
• Use the Hodge decomposition and the Hodge *-operator to define a Hermitian structure on the de Rham complex, enabling the construction of a spectral sequence converging to the cohomology of the infinite cyclic cover.
• Analyze the spectral sequence E^r_{p,q} associated with the filtration by the degree of the form, showing that E^3 is semi-simple as an R-module, leading to the degeneration of the spectral sequence at E^3.
• Apply the structure theorem for Alexander modules and the Galois invariance of Jordan block sizes to reduce the general case to the case λ = 1, where the bound is proven via Hodge-theoretic arguments.

Experimental results

Research questions

• RQ1 What is the structure of the monodromy action on the torsion part of the homology of an infinite cyclic cover of a compact Kähler manifold?
• RQ2 Can the classical bound on Jordan block sizes in the local Milnor monodromy be generalized to global algebraic and complex analytic settings?
• RQ3 What is the maximal size of Jordan blocks in the monodromy action on H_i(X^φ, ℂ) for a smooth quasi-projective variety X?
• RQ4 How does the Hodge-theoretic structure of the infinite cyclic cover constrain the Jordan form of the monodromy operator?
• RQ5 To what extent does the Galois action on eigenvalues preserve the Jordan block structure in the monodromy representation?

Key findings

• For compact Kähler manifolds, all Jordan blocks of the monodromy action on the torsion part of H_i(X^φ, ℂ) are of size one.
• For smooth quasi-projective varieties, the maximal size of Jordan blocks for eigenvalue λ is bounded by min{i+1, 2n−i}, where n is the complex dimension of the variety.
• When the monodromy is of pure Hodge type with weight 2, the bound improves to min{i+1, 2n−i−1}.
• The spectral sequence associated with the Hodge filtration on the infinite cyclic cover degenerates at the E^3 page, and all E^3 pages are semi-simple R-modules.
• The Jordan block structure is preserved under Galois conjugation, and transcendental eigenvalues yield trivial generalized eigenspaces.
• The cohomology of the infinite cyclic cover admits a filtration with semi-simple graded quotients, implying that the monodromy action is quasi-semisimple in a filtered sense.