[Paper Review] Cohomology jump loci of compact Kähler manifolds

Botong Wang|arXiv (Cornell University)|Mar 27, 2013

Algebraic Geometry and Number Theory15 references3 citations

TL;DR

This paper extends the theory of cohomology jump loci in representation varieties and moduli spaces of vector bundles on compact Kähler manifolds beyond the trivial representation. By introducing modules over differential graded Lie algebras (DGLA), it establishes that locally, the analytic germs of cohomology jump loci are isomorphic to resonance varieties via the exponential map, generalizing earlier results of Dimca-Papadima and providing new local constraints on the geometry of these loci at arbitrary semi-simple representations.

ABSTRACT

We apply the method of Dimca-Papadima to study the cohomology jump loci in the representation variety and the moduli space of vector bundles with vanishing chern classes for a compact Kähler manifold. We introduce modules over differential graded Lie algebra to extend results of Dimca-Papadima to a general point in the representation variety or the moduli space. We show that locally the cohomology jump loci is isomorphic to the resonance variety via the exponential map. This paper generalizes a previous result of the author.

Motivation & Objective

• To extend the study of cohomology jump loci in representation varieties beyond the trivial representation, particularly for compact Kähler manifolds.
• To develop a deformation-theoretic framework using differential graded Lie algebras (DGLA) and modules over DGLA to control local structure of cohomology jump loci.
• To establish a local isomorphism between cohomology jump loci and resonance varieties via the exponential map at any semi-simple representation.
• To provide new geometric and cohomological constraints on the fundamental groups of compact Kähler manifolds through the structure of these loci.

Proposed method

• Introduces a DGLA pair consisting of a DGLA and a module over it, where the DGLA controls deformations of the representation variety and the module controls cohomology group deformations.
• Uses the de Rham complex of the manifold and twisted complexes $\Omega^{\bullet}_{\textrm{DR}}(L_\rho)$ and $\operatorname{End}^\bullet(L_\rho)$ to model local geometry at a representation $\rho$.
• Defines the kernel $\mathcal{E}nd^{\bullet}_0(L_\rho)$ of the augmentation at the basepoint to capture local deformation data.
• Applies the Goldman-Millson theorem to show that the representation variety $\mathbf{R}(X,n)$ is locally isomorphic to a quadratic cone $C$ in $H^1(X, \operatorname{End}(L_\rho))$.
• Establishes an isomorphism between the formal scheme of the representation variety at $\rho$ and the formal scheme of flat connections in the DGLA model.
• Uses the exponential map to relate the local structure of cohomology jump loci to the resonance variety in the DGLA setting.

Experimental results

Research questions

• RQ1 How can the local structure of cohomology jump loci be described at non-trivial representations in the representation variety of a compact Kähler manifold?
• RQ2 What is the role of differential graded Lie algebras and modules over them in controlling the deformation theory of local systems and their cohomology?
• RQ3 To what extent do the cohomology jump loci at a semi-simple representation resemble the resonance variety via the exponential map?
• RQ4 Can the local geometry of absolute constructible sets in the Betti moduli space be characterized as cones under the exponential map?
• RQ5 What constraints do these local isomorphisms impose on the fundamental groups of compact Kähler manifolds?

Key findings

• At any semi-simple representation $\rho$, the analytic germ of the representation variety $\mathbf{R}(X,n)$ is isomorphic to a quadratic cone in $H^1(X, \operatorname{End}(L_\rho))$, generalizing the Goldman-Millson result.
• The cohomology jump loci $\mathcal{V}^i_r(X,n)$ are locally isomorphic to the resonance variety via the exponential map, extending the Dimca-Papadima framework beyond the trivial representation.
• For irreducible representations $\rho$, the local structure of $\mathcal{M}_{\textrm{B}}(X,n)$ at $\rho$ is isomorphic to a formal cone $C_{(0)}$, and each cohomology jump locus corresponds to a cone in this space.
• The cohomology jump loci at $\rho$ are locally quadratic singularities, as shown by Corollary 5.1, which applies to the moduli space of flat bundles.
• The results suggest that closed absolute constructible sets in the Betti moduli space should be locally conical under the exponential map, generalizing Simpson’s result on torsion translates of subtori.
• When $n=1$, the local isomorphism to resonance varieties implies that cohomology jump loci are unions of torsion translates of subtori, recovering a known result via new methods.