[Paper Review] Betti numbers of the moduli space of rank 3 parabolic Higgs bundles

Oscar Garcı́a-Prada, Peter B. Gothen+1 more|arXiv (Cornell University)|Nov 10, 2004

Algebraic Geometry and Number Theory28 references4 citations

TL;DR

This paper computes the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles with fixed and non-fixed determinant on a genus $ g $ Riemann surface using Morse theory. By analyzing critical submanifolds—identified as moduli spaces of parabolic triples and symmetric products—it derives explicit formulas for the Poincaré polynomial, proves the parabolic version of Laumon's theorem on the nilpotent cone being Lagrangian, and establishes the Euler characteristic vanishes. The result supports a conjecture by Hausel for general rank and provides the variant cohomology part under the action of $ \Gamma_3 $.

ABSTRACT

We compute the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles, using Morse theory. A key point is that certain critical submanifolds of the Morse function can be identified with moduli spaces of parabolic triples. These moduli spaces come in families depending on a real parameter and we study their variation with this parameter.

Motivation & Objective

• Compute the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles with fixed and non-fixed determinant on a genus $ g $ Riemann surface.
• Extend Morse-theoretic techniques from the non-parabolic case to the parabolic setting, identifying critical submanifolds as moduli spaces of parabolic triples and symmetric products.
• Prove the parabolic version of Laumon's theorem, showing the nilpotent cone is a Lagrangian subvariety in the moduli space.
• Derive the Poincaré polynomial of the moduli space and compute the variant cohomology under the action of $ \Gamma_3 $, relevant for mirror symmetry.
• Provide evidence for Hausel's conjecture on Betti numbers of parabolic Higgs bundle moduli spaces by verifying agreement in tested cases.

Proposed method

• Apply Morse theory to the $ L^2 $-norm of the Higgs field as a perfect Bott–Morse function on the moduli space.
• Identify critical submanifolds as moduli spaces of parabolic triples, which vary with a real parameter, and analyze their topological changes via this parameter.
• Use the structure of the moduli space of parabolic bundles and symmetric products of the Riemann surface to describe remaining critical submanifolds.
• Employ MacDonald's formula for the Euler characteristic of symmetric products to compute contributions from critical submanifolds of type $ (1,1,1) $.
• Combine contributions from all critical submanifolds—of types $ (1,2) $, $ (2,1) $, $ (3) $, and $ (1,1,1) $—to compute the total Poincaré polynomial.
• Use the action of $ \Gamma_3 $ on cohomology to isolate the variant part of the cohomology, leading to a formula for $ P^\mathrm{var}_t(\mathcal{M}^\Lambda) $.

Experimental results

Research questions

• RQ1 What are the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles with fixed determinant on a genus $ g $ Riemann surface?
• RQ2 How does the topology of moduli spaces of parabolic triples vary with a real parameter, and how does this affect the Morse-theoretic decomposition?
• RQ3 Is the nilpotent cone in the moduli space of parabolic Higgs bundles a Lagrangian subvariety, as in the non-parabolic case?
• RQ4 What is the Poincaré polynomial of the moduli space of rank 3 parabolic Higgs bundles with fixed determinant?
• RQ5 What is the variant part of the rational cohomology of the moduli space under the action of $ \Gamma_3 $, and how does it relate to mirror symmetry?

Key findings

• The Poincaré polynomial of the moduli space of rank 3 parabolic Higgs bundles with fixed determinant $ \Lambda $ is explicitly computed in Theorem 12.20, combining contributions from multiple critical submanifolds.
• The Euler characteristic of the moduli space of rank 3 parabolic Higgs bundles with fixed determinant is zero, as shown in Corollary 12.21.
• The variant part of the rational cohomology under $ \Gamma_3 $ has Poincaré polynomial $ P^\mathrm{var}_t(\mathcal{M}^\Lambda) = 2 \cdot 6^{n-1}(3^{2g}-1)t^{12g-12+6n}(t+1)^{4g-4} $, as stated in Theorem 12.22.
• The moduli space of parabolic bundles of rank 3 with fixed determinant has Poincaré polynomial given by Proposition 12.18, involving terms in $ t^2, t^4, t^6 $, and symmetric power contributions.
• The nilpotent cone—the preimage of zero under the Hitchin map—is proven to be a Lagrangian subvariety of the moduli space of parabolic Higgs bundles, extending Laumon's theorem to the parabolic case.
• The computed Betti numbers agree with Hausel's conjecture for parabolic Higgs bundle Betti numbers, providing strong support for the conjecture in the rank 3 case.