[Paper Review] Torus fixed points of moduli spaces of stable bundles of rank three

Thorsten Weist|ArXiv.org|Mar 4, 2009

Algebraic Geometry and Number Theory10 references3 citations

TL;DR

This paper computes the generating function for the Euler characteristic of moduli spaces of stable vector bundles of rank three on the projective plane via torus fixed points. Using quiver representations and stability conditions, it classifies fixed points by Chern classes and discriminant, showing moduli spaces are projective lines; the key result is a closed-form generating function for Euler characteristics, with explicit coefficients derived from combinatorial configurations of subspaces and filtrations.

ABSTRACT

By a result of Klyachko the Euler characteristic of moduli spaces of stable bundles of rank two on the projective plane is determined. Using similar methods we extend this result to bundles of rank three. The fixed point components correspond to moduli spaces of the subspace quiver. Moreover, the stability condition is given by a certain system of linear inequalities so that the generating function of the Euler characteristic can be determined explicitly.

Motivation & Objective

• To extend the computation of Euler characteristics of moduli spaces from rank 2 to rank 3 vector bundles on the projective plane.
• To classify torus fixed points in moduli spaces of stable bundles of rank three using filtrations of 3-dimensional vector spaces.
• To determine the structure of moduli spaces of semistable and stable representations via the subspace quiver and stability conditions.
• To derive a generating function for the Euler characteristic of these moduli spaces, based on combinatorial configurations of subspaces and Chern classes.
• To show that moduli spaces of semistable representations are isomorphic to projective lines and that fixed point components are finite and computable.

Proposed method

• Models rank 3 toric bundles as filtrations of a 3-dimensional vector space, interpreted as representations of the subspace quiver.
• Applies stability conditions via slope functions defined on dimension vectors, reducing to systems of linear inequalities.
• Uses the Euler form and GIT quotient construction to define moduli spaces of semistable and stable representations.
• Classifies fixed points by analyzing how many 2-dimensional subspaces contain given 1-dimensional subspaces, determining second Chern class and discriminant.
• Computes the generating function F(x) by summing over configurations of non-zero entries in dimension vectors, with corrections for inclusion types.
• Derives explicit formulas for discriminant D(k), D¹(k), D²(k), D³(k) to encode contributions from different inclusion patterns.

Experimental results

Research questions

• RQ1 How do torus fixed points in moduli spaces of stable bundles of rank three on P² decompose under the action of the torus?
• RQ2 What is the structure of the moduli space of semistable representations of the subspace quiver for rank 3 bundles?
• RQ3 How does the second Chern class vary across different configurations of subspaces in the filtration?
• RQ4 What is the generating function for the Euler characteristic of the moduli spaces of stable bundles of rank three on P²?
• RQ5 Under what conditions do polystable versus stable points arise, and how do they contribute to the Euler characteristic?

Key findings

• The moduli space of semistable representations of the subspace quiver for rank 3 bundles is isomorphic to the projective line P¹.
• The discriminant of stable bundles of rank three on P² is either 0 or 4 modulo 6.
• For discriminant 0, polystable points exist and their number depends only on the arm lengths of the subspace quiver.
• For discriminant 4 mod 6, only stable points exist, and the moduli space is P¹ minus two points.
• The generating function F(x) for the Euler characteristic is explicitly computed as a sum over 14 terms involving D(k), D¹(k), D²(k), D³(k), with coefficients from combinatorial types of inclusions.
• The first non-zero term in the series expansion of F(x) is −x⁻¹⁸, followed by −6x⁻³⁰, −3x⁻⁴², and −12x⁻⁴⁸, with higher-order terms showing increasing complexity and growth.