Complex Singularity Theory and Cauchy-Riemann Geometry

This book presents a rigorous, self-contained, and systematic study of Cauchy–Riemann (CR) manifolds and their deep connections to singularity theory. It synthesizes foundational contributions to CR geometry and complex singularity theory, including the solution of the complex Plateau problem and the Mather–Yau theorem, which links complex geometry with finite-dimensional commutative algebras and the Bergman function theory developed by Stephen S.-T. Yau. The discussion brings together techniques from differential geometry, several complex variables, and singularity theory within a unified framework. Complete proofs enhance the book’s value as an authoritative reference, while the concise exposition facilitates access to advanced material. Readers are expected to have a solid background in several complex variables and some familiarity with normal isolated singularities and covering spaces. The book will be of interest to a broad audience of graduate students and researchers working in the areas and topics it addresses, and much of the material may also serve as the basis for graduate-level courses.

Chapter 1 reviews the basic theory of CR geometry, including the Levi form, CR-holomorphic vector bundles, Kohn–Rossi cohomology, Hodge theory on CR manifolds, and a proof of the Boutet de Monvel theorem on the global embedding of CR manifolds.Chapter 2covers singularities, including resolution of singularities and the geometric genus.Chapter 3describes the relationship between CR invariants of strongly pseudoconvex manifolds and the invariants of interior normal isolated singularities of varieties bounded by such manifolds.Chapter 4studies the rigidity of CR morphisms using techniques from singularity theory. The final chapter presents the Bergman function theory developed by Stephen S.-T. Yau, which is used to construct explicitly infinite-dimensional moduli spaces of certain CR manifolds.

Stephen Shing-Toung Yau (Guggenheim Fellow; Life Fellow, IEEE; AMS Fellow) received the Ph.D. degree in mathematics from the State University of New York, Stony Brook, NY, USA, in 1976. He was a Member of the Institute of Advanced Study, Princeton, NJ, USA, from 1976 to 1977 and 1981 to 1982. He was a Benjamin Pierce Assistant Professor with Harvard University, Cambridge, MA, USA, from 1977 to 1980. He then joined the Department of Mathematics, Statistics and Computer Science (MSCS), University of Illinois at Chicago (UIC), Chicago, IL, USA, and served for more than 30 years. From 2005 to 2011, he was a University Distinguished Professor of UIC and Joint Professor with the Department of Electrical and Computer Engineering, MSCS, UIC. After retiring in 2011, he joined the Department of Mathematical Sciences at Tsinghua University in Beijing, China, where he served for over 10 years. His research interests include nonlinear filtering, bioinformatics, complex algebraic geometry, Cauchy–Riemann geometry, and singularities theory. Dr. Yau has been the Managing Editor and Founder of Journal of Algebraic Geometry since 1991 and the Editor-in-Chief and Founder of Communications in Information and Systems since 2000. He was the General Chairman of the 1995 IEEE International Conference on Control and Information. He received the Sloan Fellowship in 1980, the Guggenheim Fellowship in 2000, and the American Mathematical Society Fellow Award in 2013. In 2005, he was entitled the UIC Distinguished Professor. In 2019, he won the Chern Prize of Lifetime Achievement in Mathematics.

Bingui Chen is professor of mathematics at Sun Yat-sen University in Guangzhou, China.

Xiankui Meng is progressor of mathematics at Beijing University of Posts and Telecom in Beijing, China.

Huaiqing Zuo is professor of mathematics at Tsinghua Universit in Beijing, China.