Transversality Methods for Homotopy Groups of Stable Loci in Affine GIT Quotients

TL;DR

This paper extends transversality methods to study the homotopy groups of stable loci in affine GIT quotients, providing new topological insights into stability conditions across various fields.

Contribution

It develops a general GIT framework using infinite-dimensional transversality, applicable even when semistability differs from stability, and illustrates this with diverse examples.

Findings

Matches known topology of the 2-Kronecker quiver stable locus

Determines connectivity of controllable systems in control theory

Computes the topology of data samples with unique MLE in statistical models

Abstract

We investigate the homotopy groups of stable loci in affine Geometric Invariant Theory (GIT), arising from linear actions of complex reductive algebraic groups on complex affine spaces. Our approach extends the infinite-dimensional transversality framework of Daskalopoulos-Uhlenbeck and Wilkin to this general GIT setting. Central to our method is the construction of a G-equivariant holomorphic vector bundle over the conjugation orbit of a one-parameter subgroup (1-PS), whose fibres are precisely the negative weight spaces determining instability. A key proposition establishes that a naturally defined evaluation map is transverse to the zero section of this bundle, implying that generic homotopies avoid all unstable and strictly semistable strata under certain dimensional inequalities. Our result also covers cases where semistability does not coincide with stability. The applicability of this framework is illustrated by several examples. For the 2-Kronecker quiver, our conclusions precisely match the known topology of the stable locus. In linear control theory, where GIT stability corresponds to the notion of controllability, our results determine the connectivity of the space of controllable systems. In statistical modelling, where stability for star-shaped Gaussian model corresponds to the existence of a unique Maximum Likelihood Estimate, we compute the connectivity of the space of data samples that yield such a unique estimate, providing topological insight into the problem of parameter non-identifiability.