[Paper Review] Co-Higgs bundles on P^1

Steven Rayan|arXiv (Cornell University)|Oct 12, 2010

Algebraic Geometry and Number Theory22 references3 citations

TL;DR

This paper classifies co-Higgs bundles on the projective line 𝔼¹ by their Grothendieck splitting types, establishing necessary and sufficient conditions for the existence of stable Higgs fields. It explicitly describes the rank-2, odd-degree moduli space as a universal elliptic curve via a globally defined algebraic equation, verifies conjectural Betti numbers for ranks 2–5 using Morse theory and ADHM recursion, and conjectures degree-invariant Betti numbers for higher ranks.

ABSTRACT

Co-Higgs bundles are Higgs bundles in the sense of Simpson, but with Higgs fields that take values in the tangent bundle instead of the cotangent bundle. Given a vector bundle on P^1, we find necessary and sufficient conditions on its Grothendieck splitting for it to admit a stable Higgs field. We characterize the rank-2, odd-degree moduli space as a universal elliptic curve with a globally-defined equation. For ranks r=2,3,4, we explicitly verify the conjectural Betti numbers emerging from the recent work of Chuang, Diaconescu, Pan, and Mozgovoy on the ADHM formula. We state the result for r=5.

Motivation & Objective

• To characterize vector bundles on ℙ¹ admitting stable co-Higgs fields via their splitting types.
• To provide a global algebraic description of the rank-2, odd-degree moduli space of co-Higgs bundles on ℙ¹.
• To verify conjectural Betti numbers for co-Higgs moduli spaces of ranks 2–5 using Morse theory and ADHM recursion.
• To investigate degree independence of Betti numbers in co-Higgs moduli spaces on ℙ¹.
• To extend and confirm conjectures on Hodge polynomials and Poincaré polynomials from ADHM formulae in genus 0.

Proposed method

• Uses Grothendieck's splitting theorem to classify holomorphic vector bundles on ℙ¹ by their splitting types.
• Applies slope-stability conditions to co-Higgs bundles with Higgs fields valued in the tangent bundle (K*), ensuring φ ∧ φ = 0 is automatic on curves.
• Employs Morse theory on the Hitchin fibration to compute Betti numbers for ranks 3 and 4, using fixed-point loci and chain decompositions.
• Utilizes the ADHM recursion formula from Chuang, Diaconescu, Pan, and Mozgovoy to verify conjectural Poincaré polynomials for ranks 2–5.
• Constructs holomorphic chains from Jordan–Hölder filtrations to analyze moduli spaces and compute Betti numbers via Morse index data.
• Applies degree duality to relate moduli spaces of degree d and −d, reducing computations for negative degrees.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions on the splitting type of a vector bundle on ℙ¹ for it to admit a stable co-Higgs field?
• RQ2 Can the rank-2, odd-degree moduli space of co-Higgs bundles on ℙ¹ be described globally by an algebraic equation?
• RQ3 Do the Betti numbers of co-Higgs moduli spaces on ℙ¹ match the conjectural Poincaré polynomials derived from the ADHM recursion formula?
• RQ4 Is the Betti number structure of co-Higgs moduli spaces independent of the degree of the bundle?
• RQ5 How do holomorphic chain decompositions and Morse theory facilitate the computation of Betti numbers in higher-rank cases?

Key findings

• The rank-2, odd-degree moduli space of co-Higgs bundles on ℙ¹ is a smooth, globally defined universal elliptic curve given by a single algebraic equation.
• The Betti numbers of the rank-2, odd-degree moduli space are those of S², confirming a topological characterization.
• For rank 3 and rank 4 with odd degree, the Betti numbers computed via Morse theory match the conjectural Poincaré polynomials from the ADHM formula.
• The Poincaré polynomial for rank 5 and degree −1 is explicitly computed as 1 + x² + 3x⁴ + 5x⁶ + 10x⁸ + 15x¹⁰ + 26x¹² + 38x¹⁴ + 56x¹⁶ + 77x¹⁸ + 105x²⁰ + 131x²² + 156x²⁴ + 165x²⁶ + 154x²⁸ + 103x³⁰ + 40x³².
• The paper conjectures that Betti numbers of co-Higgs moduli spaces on ℙ¹ are independent of degree, based on the absence of degree parameters in the ADHM conjectural Poincaré polynomials.