A Transversality Method for Homotopy Groups of Stable Loci in Affine GIT

This thesis studies the homotopy groups of stable loci arising in affine Geometric Invariant Theory (GIT). We begin by establishing the foundational theory of reductive linear algebraic groups and their actions on affine varieties. In particular, we review Nagata's theorem on the finite generation of invariant rings.

The main contribution of this work is to extend the transversality framework of Daskalopoulos--Uhlenbeck and Wilkin to the affine setting. We construct a $G$-equivariant holomorphic vector bundle $\pi : W \to G\cdot \lambda$ over the conjugation orbit of a 1-PS, where the fibres are negative weight spaces $V(\lambda)_-$. We prove that the evaluation map $D$ is transverse to the zero section of this bundle. Consequently, we establish that for generic homotopies, the preimage of the zero section is empty under specific dimensional inequalities, i.e. these homotopies avoid non-stable strata.

Our main theorem asserts that the stable locus $V^{st}$ is $(d_{min}-2)$-connected, where $d_{min}$ is the minimal real codimension of the unstable strata. The applications of this result are demonstrated by several examples including the $2$-Kronecker quiver, the space of controllable linear systems, and the uniqueness of Maximum Likelihood Estimates for Gaussian graphical models.