[GS_M_AG] The Wahl defect of normal surface singularities: positivity and unbounded negativity
ABSTRACT
The classical Milnor--Tjurina inequality for an isolated surface singularity that is a hypersurface states that $\mu \geq \tau$, where $\mu=b_2(M)$ is the Milnor number and $\tau$ is the dimension of the Tjurina algebra; equality holds precisely in the quasi-homogeneous case.
Given a minimal good resolution $f:(X,E) \to (V,0)$, set $S_X=\Theta_X(-\log{E})$ and define the Wahl defect by $\Delta(V,0)=h^1(X,\mathcal{O}_X)-h^1(X,S_X)+h^1(X,\bigwedge^2 S_X)$. For a smoothing $\pi$, Wahl's formula gives $\Delta(V,0)=1+\mu_\pi-\tau_\pi+\alpha_\pi$, where $\mu_\pi=b_2(M_\pi)$, $\tau_\pi$ is the dimension of the corresponding ordinary smoothing component, and $\alpha_\pi$ is a correction associated with the relative dualizing sheaf.
As a natural non-Gorenstein extension of the hypersurface phenomenon, Wahl's Main Conjecture proposed that $\Delta(V,0) \geq 0$, with equality precisely for singularities admitting a good $\mathbb{C}^*$-action. The Main Conjecture has two principal specializations: the Rational Conjecture assumes that $(V,0)$ is rational, so that $h^1(X,\mathcal{O}_X)=0$, whereas the $\mathbb{Q}$-Gorenstein Conjecture assumes that $\pi$ is $\mathbb{Q}$-Gorenstein, so that $\alpha_\pi=0$.
We prove both the Rational and $\mathbb{Q}$-Gorenstein Conjectures, including their equality characterizations. By contrast, we construct an infinite family of non-$\mathbb{Q}$-Gorenstein singularities with $\Delta(V,0) \to -\infty$, thereby disproving the unrestricted Main Conjecture.
As applications, for rational surface singularities, our theorems give graph-theoretic upper bounds for equisingular analytic moduli and for the dimension of the Artin component, with equality in either bound characterizing quasi-homogeneity. Our results also complete the analytic classification of normal surface singularities admitting rational homology disk smoothings.
This is joint work with Heesang Park.
