[Paper Review] Moduli spaces of bundles over Riemann surfaces and the Yang-Mills stratification revisited

Frances Kirwan|ArXiv.org|May 24, 2003

Algebraic Geometry and Number Theory25 references7 citations

TL;DR

This paper refines the Yang-Mills stratification of the space of connections over a Riemann surface by introducing three new filtrations—maximal and minimal Jordan-Hölder filtrations, and a balanced δ-filtration of maximal triviality—each yielding a stratification of the space of connections by locally closed complex submanifolds of finite codimension. The key contribution is establishing that these refinements provide a framework for understanding the cohomology relations of moduli spaces of stable bundles when rank and degree are coprime and rank exceeds two.

ABSTRACT

Refinements of the Yang-Mills stratifications of spaces of connections over a compact Riemann surface are investigated. The motivation for this study was the search for a complete set of relations between the standard generators for the cohomology of the moduli spaces of stable holomorphic bundles of rank n and degree d when n and d are coprime and n>2.

Motivation & Objective

• To address the lack of a complete set of relations among standard cohomology generators for moduli spaces M(n,d) when n > 2 and gcd(n,d) = 1.
• To extend the understanding of cohomology relations beyond the n=2 case, where Mumford's conjecture was proven.
• To develop refined stratifications of the space of connections using Jordan-Hölder and δ-filtrations to model the structure of semistable and stable bundles.
• To relate these refined stratifications to the Harder-Narasimhan filtration and the Yang-Mills functional via moment map theory.
• To provide a geometric framework for investigating the cohomology ring structure of M(n,d) through Morse-theoretic and geometric invariant theory techniques.

Proposed method

• Uses the Yang-Mills functional as a norm-square of a moment map to define a Morse stratification on the space of connections C(n,d).
• Applies the refinement method from Kirwan (1985) to the Yang-Mills stratification, producing new stratifications with stable bundles as the open stratum.
• Introduces two canonical refinements of the Harder-Narasimhan filtration: the maximal and minimal Jordan-Hölder filtrations.
• Defines a third refinement via the balanced δ-filtration of maximal triviality, constructed using subquotients of the Harder-Narasimhan filtration.
• Applies the stratification to subvarieties appearing in the inductive construction of the Yang-Mills stratification, using projectivized normal bundles.
• Uses the indexing set of the stratification to analyze the structure of strata and their codimensions, particularly in relation to the cohomology of M(n,d).

Experimental results

Research questions

• RQ1 How can the Yang-Mills stratification of the space of connections be refined to better understand the cohomology of moduli spaces of stable bundles?
• RQ2 What are the geometric and algebraic properties of the maximal and minimal Jordan-Hölder filtrations of semistable bundles?
• RQ3 How does the balanced δ-filtration of maximal triviality relate to the Harder-Narasimhan and Jordan-Hölder filtrations?
• RQ4 Can these refined filtrations be used to construct a complete set of relations among the standard cohomology generators of M(n,d) for n > 2?
• RQ5 What is the relationship between the three refinements—maximal JH, minimal JH, and balanced δ-filtration—and how do they differ in structure?

Key findings

• Three distinct refinements of the Harder-Narasimhan filtration are constructed: maximal Jordan-Hölder, minimal Jordan-Hölder, and balanced δ-filtration of maximal triviality.
• Each refinement induces a stratification of the space of connections C(n,d) by locally closed complex submanifolds of finite codimension, with the stable connections as the open stratum.
• The maximal and minimal Jordan-Hölder filtrations of a direct sum of semistable bundles are not necessarily refinements of each other, and differ in their extension rules at the ends.
• The balanced δ-filtration of maximal triviality is shown to be a refinement with no proper refinement preserving subquotients of the same slope, and is characterized by minimizing a certain error term ε.
• The three filtrations are related via the concept of a pivotal filtration, and their interplay is formalized in Theorem 7.11 and Propositions 7.4, 7.6, and 7.8.
• The refined stratifications provide a geometric framework for studying cohomology relations in M(n,d), particularly for n > 2, where the standard generalization of Mumford’s relations fails.