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

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. 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. We also consider Helmke systems and show that for stability parameters satisfying certain bounds, the space of systems that are both controllable and observable is exactly the space of stable points. The main result can then be used to compute the connectivity of this space.