Pure and Applied Mathematics Quarterly

Volume 18 (2022)

Number 2

Special issue in honor of Joseph J. Kohn on the occasion of his 90th birthday

Guest Editors: J.E. Fornaess, Stanislaw Janeczko, Duong H. Phong, and Stephen S.T. Yau

Kohn–Rossi cohomology and the complex Plateau problem

Pages: 683 – 718

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n2.a15

Authors

Xiankui Meng (School of Science, Beijing University of Posts and Telecommunications, Beijing, China)

Stephen Shing-Toung Yau (Department of Mathematical Sciences, Tsinghua University, Beijing, China; and Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing, China)

Abstract

In the present work, we investigate the relationship between compact strongly pseudoconvex CR manifolds and the singularities of their Stein fillings. We compute the dimensions of Kohn–Rossi cohomology groups with values in holomorphic vector bundles in terms of local cohomology groups. As an application, we solve the classical complex Plateau problem for compact strongly pseudoconvex CR manifold $X$ when its Stein filling $V$ has only isolated complete intersection singularities. This generalizes earlier results of Yau.

Keywords

Kohn–Rossi cohomology, local cohomology, isolated complete intersection singularity, projective resolution

2010 Mathematics Subject Classification

Primary 32S05, 32S10, 32V05, 32V15. Secondary 32E10, 32T15.

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The first author is supported by the National Natural Science Foundation of China (grant no. 11901046) and the National Key Research and Development Program of China (No. 2021YFA1002600).

The second author is supported by the National Natural Science Foundation of China (grant nos. 11961141005), Tsinghua University start-up fund, and Tsinghua University Education Foundation fund (042202008).

Received 6 November 2021

Accepted 29 December 2020

Published 13 May 2022