[GS_M_AG] Relative Equisingular Deformation Theory under Resolution Subgraph Surgery
ABSTRACT
This talk develops a relative approach to equisingular deformation theory for rational surface singularities, viewed through surgery operations on connected subgraphs of a resolution graph. The central problem is to compare the logarithmic tangent cohomology $H^1(V, T_V(-\log{E}))$ before and after removing or attaching a terminal linear chain, with the aim of obtaining effective estimates for $h^1(V, T_V(-\log{E}))$. In the terminal-linear-chain case, we describe how the self-intersection data, the infinitesimal motion of the attaching points, and the analytic gluing contribution are reflected in the resulting variation of logarithmic cohomology. This relative viewpoint is intended to separate the graph-theoretic contribution from the analytic-moduli-dependent contribution, thereby providing a deformation-theoretic calculus along the resolution graph. As applications, we discuss possible implications of these \(h^1\)-estimates for Wahl's Rational Conjecture, the classification of rational surface singularities admitting rational homology disk smoothings, and related questions in the deformation theory of surface singularities; part of this work in progress is joint with Márton Beke, Heesang Park, András Stipsicz.
