Positivity of Wahl defects and representations from index-one covers

Source abstract

Let $(X,E) \to (V,0)$ be the minimal good resolution of a normal complex surface singularity, with $E$ reduced. The Wahl defect is $Δ(V,0) :=h^1(\mathcal{O}_X)-h^1(S_X)+h^1(\bigwedge^2 S_X)$, where $S_X=Θ_X(-\log E)$. For non-Gorenstein $\mathbb{Q}$-Gorenstein singularities $(V,0)$, we prove that $Δ(V,0)\geq0$, with equality precisely in the quasi-homogeneous case. This proves Wahl's Main Conjecture in this class and his $\mathbb{Q}$-Gorenstein and Rational Conjectures. The proof expresses $Δ$ as the multiplicity of the canonical character in a representation constructed from differential forms and residues. We also prove $Δ+4p_g\geq0$ for arbitrary normal surface singularities, with equality precisely for rational quasi-homogeneous singularities. For smoothings of rational surface singularities, we give a Riemann--Roch formula for the lengths of the cokernels of the restriction maps to the central fiber for all reflexive canonical powers, in terms of the Milnor number and periodic local Riemann--Roch corrections. This yields numerical and representation-theoretic criteria for $\mathbb{Q}$-Gorenstein smoothings, defined as quotients of equivariant smoothings of the index-one cover with trivial group action on the base. If a rational singularity admits a $\mathbb{Q}$-Gorenstein smoothing, such smoothings are precisely those with minimal Milnor number, or equivalently those whose smoothing components have minimal dimension. Combining these results, we prove that every rational homology disk smoothing is $\mathbb{Q}$-Gorenstein, has a rational weighted homogeneous central fiber, and occurs over a one-dimensional smoothing component. Together with the known weighted homogeneous classification, this completes the analytic classification of normal surface singularities admitting rational homology disk smoothings.