Methods and Applications of Analysis

Volume 24 (2017)

Number 2

Special issue dedicated to Henry B. Laufer on the occasion of his 70th birthday: Part 2

Guest Editors: Stephen S.-T. Yau (Tsinghua University, China); Gert-Martin Greuel (University of Kaiserslautern, Germany); Jonathan Wahl (University of North Carolina, USA); Rong Du (East China Normal University, China); Yun Gao (Shanghai Jiao Tong University, China); and Huaiqing Zuo (Tsinghua University, China)

An intrinsic approach to stable embedding of normal surface deformations

Pages: 277 – 292

DOI: https://dx.doi.org/10.4310/MAA.2017.v24.n2.a4

Author

Adam Harris (School of Science and Technology, University of New England, Armidale, NSW, Australia)

Abstract

We introduce the notion of involutive Kodaira–Spencer deformations of the regular part $X_0$ of a normal surface singularity, which form a subspace of the analytic cohomology $H^1 (X_0, T^{1,0} X_0)$. Examples of involutive deformations for which the Stein completion does not embed in a complex Euclidean space of stable dimension are in fact well-known. Under the assumption that $X_0$ admits a Kähler metric with $L^2$-curvature, we show that unstable deformations are avoided if the holomorphic functions which determine an embedding of the central fibre are correspondingly deformed into functions which can be uniformly bounded on compact subsets.

Keywords

singularities, Kodaira–Spencer deformations, holomorphic embedding

2010 Mathematics Subject Classification

32G05, 32S30

Full Text (PDF format)

Received 31 October 2016

Accepted 27 March 2017

Published 3 January 2018