New invariants for complex manifolds and rational singularities
In the article under review, the authors study two invariants \(f^{(1,1)}\) and \(g^{(1,1)}\) for rational surface singularities. If \((M,A)\to (V,0)\) is a resolution of a normal surface singularity \((V,0)\) with exceptional set \(A\), then \(f^{(1,1)}=\dim \Gamma(\Omega_M^2)/\langle \Gamma(\Omega_M^1) \wedge \Gamma(\Omega_M^1) \rangle\) and \(g^{(1,1)}\) is defined in a similar way on \(M\setminus A\). Yau conjectured that these invariants are strictly positive for all normal surface singularities, and \textit{R. Du} and \textit{S. Yau} [Commun. Anal. Geom. 18, No. 2, 365--374 (2010; Zbl 1216.32018); J. Differ. Geom. 90, No. 2, 251--266 (2012; Zbl 1254.32051)] confirmed this conjecture for weighted homogeneous singularities, rational double points, and cyclic quotient singularities. In this paper, the authors prove that \(f^{(1,1)}= g^{(1,1)}\geq 1\). As an application, they solve the regularity problem of the Harvey-Lawson solution to the complex Plateau problem for a strongly pseudoconvex compact rational CR manifold of dimension three. The second half of the article devoted to the explicit calculation to show \(f^{(1,1)}= g^{(1,1)}= 1\) for rational triple points using local coordinates on the minimal resolution space.
scientific article; zbMATH DE number 6382630