Research Seminar "Analysis of Several Complex Variables"

2020/2021

Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'

Category 'Best Course for New Knowledge and Skills'

Type: Optional course (faculty)

Delivered by: Faculty of Mathematics

Where: Faculty of Mathematics

When: 3, 4 module

Instructors: Alexey Glutsyuk

Language: English

ECTS credits: 3

Contact hours: 36

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Abstract

Analysis of several complex variables is not studied in the usual university program. At the same time it is a necessary pre-requisite to study many important domains of contemporary mathematics such as algebraic geometry, complex dynamics, singularity theory, differential equations etc. While holomorphic functions of several complex variables share many basic properties of functions of one variable, new phenomena of analytic extention occurs. For example, they can have neither isolated singularities, nor compact sets of singularities. Each complex space of dimension at least two contains a proper domain that is biholomorphically equivalent to the ambient space (Fatou——Bieberbach domain). Theory of holomorphic convexity and Stein manifolds together with basic sheave theory allow to prove important extension and approximation theorems. The GAGA principle in algebraic geometry says that every analytic object on a complex projective algebraic manifold is algebraic. The cours will cover the above mentioned topics, including basic analytic set theory, biholomorphic automorphisms and introduction to complex dynamics.

Learning Objectives

To make students familiar with the main basic theorems of multidimensional complex analysis, e.g., Hartogs' theorem, theorem on erasing compact (or codimension >1) singularities.
To study analytic set theory, Wejerstrtass polynomials, branched coverings, explain Proper Mapping Theorem and its importance by providing applications, e.g., proof of Chow theorem in algebraic geometry.
To introduce intersection index and show that each automorphism of a projective space is a projective transformation.
To study automorphism groups of polydisc and ball.
To make an introduction to complex dynamics, demonstrate important interplay between complex analysis and dynamics by showing how the dynamics produces Fatou—Bieberbach domains: exotic copies of affine complex spaces.
To study the notions of domain of holomorphy, holomorphic convexity, pseudoconvexity, Levi convexity and to make clear relations between them and equivalence of their different definitions.
To introduce Levi form and formulate conditions (one necessary, another sufficient) for a domain with smooth boundary to be Levi convex. To make clear that Levi-positive hypersurfaces admit a natural contact structure.
To introduce Dolbeault ahd sheaf cohomology, prove extension and approximation theorems for functions on analytic subsets in holomorphically convex.
To introduce Stein manifolds and make clear equivalence of their different definitions (including their realization as submanifolds in affine complex spaces). To introduce coherent sheaves and study Cartan's A and B theorems. Extend the above-mentioned extension and approximation theorem to Stein manifolds.

Expected Learning Outcomes

Know equivalent definitions of holomorphic function, Cauchy—Riemann equations. Know proofs of Osgood Lemma on separate holomorphicity and Cauchy formula. Know definition of convergence domain of Taylor series, proof of its rotation-invariance and be able to find it for simplest examples. Know proofs of Hartogs Erasing Singularity Theorem and the theorems on erasing compact and complex codimension >1 singularities.
Know definitions and main properties of analytic sets and Weierstrass polynomials. Proof of Weierstrass Preparation Theorem. Know definitions of irreducible analytic sets and irreducible germs of analytic sets. Know proofs of factoriality of local ring of germs of holomorphic function and of geometric criterium of irreducibility in terms of zero locus.
Know criterium of irreducibility of germ of function in two variables (in terms of parametrization) and its proof. Solve exercises to check irreducibility of simplest germs. Know the general statement of Remmert Proper Mapping Theorem. Know definition and basic properties of projective algebraic sets and proof of Chow Theorem. Solve exercises to check properness of simplest maps.
Know statements and proofs of Generalized Maximum Principle, multidimensional Schwarz Lemma and Cartan automorphism theorem. Know description of automorphisms of polydisk and ball. Solve exercise to find an automorphism sending a given point to a given point. Know statement and poof of linearization theorem for germs of biholomorphic maps. Know the definition and construction of Fatou--Bieberbach domain with proof.
Know definitions of domain of holomorphy and holomorphically convex domains and proof of their equivalence. Proof of log-convexity criterium for being a convergence domain of a Taylor series
Know what does it mean holomorphic non-extendablility of domains. Proof of sufficient condition: Continuity Principle. Know what is Levi-convexity and how is it related to holomorphic convexity. Know statement and proof of sufficient condition for Levi-convexity in term of Levi form. Know how to construct contact structure on the boundary of a Levi-convex domain. Be able to solve exercises on Levi convexity of simplest domains.
Know definition of Dolbeault cohomology and proof of Poincare Lemma. Know the definition of Cousin problems, proof of their solution using d-bar problem and corollaries for analytic hypersurfaces (sufficient condition for existence of defining function). Know definition of sheafs and sheaf cohomology, exact sequences; using them, prove extension theorem for functions on submanifold.
Know equivalent definitions and basic properties of Stein manifolds. Know definitions of coherent analytic sheaves and basic examples. Know proof of extension and approximation theorems on Stein manifolds and statements of Cartan A and B Theorems (without proofs).

Course Contents

Basic properties of holomorphic functions of several complex variables. Hartogs and other erasing singularity theorems.
Analytic sets; definition, main properties, Erasing analytic singularities of bounded functions. Weierstrass polynomials and preparatory theorem. Factoriality of local ring of germs of holomorphic functions. Characterization of irreducible germs
Stein manifolds, coherent analytic sheaves, extension and approximation theorems, Cartan A and B Theorems.
Dolbeault cohomology, Poincare lemma, Cousin problems, Sheaf cohomology.
Zero loci of functions of two variables: necessary irreducibility conditions and parametrization Remmert Proper Mapping Theorem. Introduction to algebraic geometry. Chow Theorem.
Holomorphic non-extendablility of domains. Sufficient condition: Continuity Principle. Levi-convexity, relation to holomorphic convexity. Levi form. Sufficient condition for Levi-convexity. Contact structure.
Domains of holomorphy and holomorphic convexity: equivalence (Oka Theorem) Characterization of convergence domains of power series as log-convex complete Reinhardt domains
Generalized Maximum Principle and Schwarz Lemma. Cauchy Inequality. Cartans theorem on automorphisms of bounded domains, automorphisms of ball and polydisk. Introduction to complex dynamics: linearization theorems, polynomial automorphisms, Fatou—Bieberbach domains.

Assessment Elements

solution of problems from problem lists
intermediate written work on theory and problems
exam

Interim Assessment

Interim assessment (4 module)
Total score (grade): 0.3 (grade for problem solving) + 0.1 (grade for the intermediate written workm theory and problems) + 0.6 (grade for exam)

Bibliography

Recommended Core Bibliography

Аналитические функции многих комплексных переменных, Ганнинг, Р., 1969
Введение в комплексный анализ. Ч. 2: Функции нескольких переменных, Шабат, Б. В., 1976
Принципы алгебраической геометрии. Т. 1: ., Гриффитс, Ф., 1982

Recommended Additional Bibliography

Комплексные аналитические множества, Чирка, Е. М., 1985

Course Syllabus