New invariants for complex manifolds and rational singularities

Rong Du and Yun Gao

Vol. 269 (2014), No. 1, 73–97

DOI: 10.2140/pjm.2014.269.73

Abstract

Two new invariants $f^{\left(1,1\right)}$ and $g^{\left(1,1\right)}$ were introduced by Du and Yau for solving the complex Plateau problem. These invariants measure in some sense how far away the complex manifolds are from having global complex coordinates. In this paper, we study these two invariants further for rational surface singularities. We prove that these two invariants never vanish for rational surface singularities, which confirms Yau’s conjecture for strict positivity of these two invariants. As an application, we solve regularity problem of the Harvey–Lawson solution to the complex Plateau problem for a strongly pseudoconvex compact rational CR manifold of dimension $3$. We also construct resolution manifolds for rational triple points by means of local coordinates and show that $f^{\left(1,1\right)}=g^{\left(1,1\right)}=1$ for rational triple points.

Dedicated to Professor Stephen S.-T. Yau on the occasion of his sixtieth birthday.

Keywords

complex Plateau problem, strongly pseudoconvex, CR manifold, rational triple points

Mathematical Subject Classification 2010

Primary: 14B05, 32V15

Secondary: 32S25, 57P05

Milestones

Received: 7 October 2011

Accepted: 21 June 2013

Published: 15 July 2014