New invariants for complex manifolds and rational singularities
type: Item
instance of: scholarly article
title: New invariants for complex manifolds and rational singularities
zbMATH Open document ID: 1311.14005
author: Rong Du
author: Yun Gao
published in: Pacific Journal of Mathematics
publication date: 2014-12-30T00:00:00Z
review text: 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.
reviewed by: Tomohiro Okuma
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 32S25
Mathematics Subject Classification ID: 32V15
zbMATH DE Number: 6382630
zbMATH Keywords: rational surface singularity
zbMATH Keywords: rational triple points
zbMATH Keywords: complex Plateau problem
zbMATH Keywords: strongly pseudoconvex
zbMATH Keywords: CR manifold
MaRDI profile type: Publication
full work available at URL: pjm.2014.269.73
OpenAlex ID: W2011924761
DOI: 10.2140/PJM.2014.269.73
description: scientific article; zbMATH DE number 6382630