Math-601D-201

Math-601D-201

Topics in analysis in several complex variables

Summary: Course Outline

Instructor:

Charles Favre
charles dot favre at polytechnique dot edu

Schedule:

Tuesday: 11.00 -- 12.30 in Room MATH 105
Thursday: 11.00 -- 12.30 in Room MATH 105
Office Hours: Tuesday & Thursday 14.00 -- 15.30 in Room LSK 300C, or by appointment

Synopsis :

This is a 3-credits course which is intended as an introduction to complex analysis in several complex variables and its interactions with complex geometry. An emphasis will be put on analytic aspects of the theory (d-bar operator, plurisubharmonic functions, currents). We shall cover the following topics.

Holomorphic functions in Cⁿ and Hartog's phenomenon;
Local geometry of analytic subsets, Oka and Cartan's coherence theorem;
Subharmonic and plurisubharmonic functions;
Pseudoconvex domains and the solution to the Levi problem;
Positive currents and applications (if times allows).

Prerequisites: a basic course in complex analysis in one variable such as Math-300:101. A basic knowledge on the notions of manifolds and differential forms is preferable but it is not necessary for the understanding of the course.

Main references for the course:

L. Hörmander. An introduction to complex analysis in several variables. ISBN-13 : 978-0444884466. The book is available online through UBC library subscription.

Some complementary references include:

J.-P. Demailly. Complex analytic and differential geometry.
R. C. Gunning. Introduction to Holomorphic Functions of Several Variables, Volume I: Function Theory & Volume II: Local Theory.
E.-M. Chirka. Complex analytic sets.
L. Kaup and B. Kaup. Holomorphic functions of several variables: an introduction to the fundamental theory

Evaluation

Grades will be based two homework problems sets (one at mid-term and one at the end of the course). Weekly homework exercises will be also assigned but not graded.
Project 1: Fatou-Bieberbach domains. See Rosay-Rudin "Holomorphic maps from Cⁿ to Cⁿ", and Stentsones Fatou-Bieberbach domains with smooth boundary
Project 2: Hodge decomposition for complex tori. See the book by Lange and Birkenhake, "Complex abelian varieties", chapter 1.

Homeworks

Lectures:

Lecture 1: January 7th
- Holomorphic functions in dimension 1: analytic and conformal maps.
- Holomorphic functions in open subsets of Cⁿ: definition.
- Homework: exercices 1 and 2 (chapter 1)
Lecture 2: January 9th
- Holomorphic functions in several complex variables: Cauchy formula.
- The analytic implicit function theorem
- Homework: exercice 3 (chapter 1)
- Notes for the lectures 1 and 2 (by S. Asgarli)
Lecture 3: January 14th
- Power series and Reinhardt domains
- Statements of Hartog's extension theorem and the resolution of the dbar equation with compact support in Cⁿ
- Homework: exercice 7 (chapter 1)
Lecture 4: January 16th
- Proof resolution of the dbar equation with compact support in Cⁿ
- Proof of Hartog's theorem
- Definition of domains of holomorphy and holomorphic hull
- Statement of the equivalence between being a domain of holomorphy and being holomorphically convex
- Homework: exercice 9 (chapter 1)
- Notes for the lectures 3 and 4 (by S. Asgarli)
Lecture 5: January 21st
- Proof of the equivalence between being a domain of holomorphy and being holomorphically convex
- Applications: image under a biholomorphism, product of domains of holomorphy, convex domains, {|f|<1}
- Statement that a Reinhardt domain is a domain of holomorphy iff it is complete and log-convex
- Homework: exercice 4 & 10 (chapter 1)
Lecture 6: January 23rd
- Proof that complete log-convex Reinhardt domain are domains of holomorphy
- domains of holomorphy can be characterized by the geometry of their boundary (statement of the Levi-Oka theorem)
- Rothstein theorem: there is no proper holomorphic map from the unit polydisk to the euclidean unit ball
- Poincaré-Cartan's approach to proving that the unit polydisk and the euclidean unit ball are not biholomorphic one to the other
- Homework: exercice 11 (chapter 1)
- Notes for the lectures 5 and 6 (by S. Asgarli)
Lecture 7: January 28th
- Complex manifolds : basics
- Complex submanifolds
- Hopf manifolds
- Homework: exercices 1 & 2 (chapter 2)
Lecture 8: January 30th
- Analytic sets : definition
- Regular and singular loci of an analytic set
- Proof of the density of the regular set
- Proof of the analycity of the singular set when defined by one equation
- Homework: exercices 4 & 5 (chapter 2)
- Notes for the lectures 7 and 8 (by S. Asgarli)
Lecture 9: February 4th
- the ring O_n of holomorphic germs at 0 in Cⁿ
- Weierstrass preparation theorem
- Homework: exercices 6 & 7 (chapter 2)
Lecture 10: February 6th
- the ring of holomorphic germs at 0 in Cⁿ is Noetherian and factorial
- germs of analytic sets and ideals of O_n
- Homework: exercices 8 (chapter 2)
- Notes for the lectures 9 and 10 (by S. Asgarli)
Lecture 11: February 11th
- decomposition of germs of analytic sets into irreducible components
- sheaves of O_V-modules
- coherent sheaves on a complex manifolds
- Oka's theorem: statement
- Homework: exercices 9, 10 (chapter 2)
Lecture 12: February 13th
- morphisms of sheaves, kernel and image sheaf of a morphism
- translation of Oka's coherence theorem into a statement P_qⁿ
- proof that P₁ⁿ and P_q-1ⁿ implies P_qⁿ
- proof that P_q^n-1 for all q implies P_q¹
- Homework: exercices 11, 12 (chapter 2)
- Notes for the lectures 11 and 12 (by S. Asgarli)
Winterbreak: February 18th and 20th
Lecture 13: February 25th
- Structure of prime ideals in O_n including Noether normalization lemma, and existence of a universal denominator
Lecture 14: February 27th
- local parameterization theorem (statement and proof)
- proof of the analytic Nullstellensatz
- The singular locus of an analytic set is analytic of strictly smaller dimension (granting Cartan's coherence theorem)
- Notes for the lectures 13 and 14 (by S. Asgarli)
Lecture 15: March 3rd
- Cartan's coherence theorem : proof of Gunning based on the coherence of quotient ideal sheaves.
- Complements: four more coherence theorems (radical ideal sheaves, structure sheaves of analytic sets, normalization sheaves, higher direct images of coherent sheaves under proper holomorphic maps).
- Riemann's extension theorems: bounded holomorphic functions through arbitrary analytic sets; and holomorphic functions through analytic subsets of codimension 2.
Lecture 16: March 5th
- Harmonic functions: definition, harmonic functions are real parts of holomorphic functions locally, mean-value equality
- Subharmonic function: definition (usc and dominated by harmnonic functions)
- Characterization in terms of submean inequalities. Maximum principle.
- Main properties: local property, stable by sum, by max, by decreasing limits, by composing with convex increasing functions
- Local integrability of subharmonic functions
- Notes for the lectures 15 and 16 (by S. Asgarli)
Lecture 17: March 10th
- regularization of subharmonic functions
- characterization of subharmonic functions in terms of their distributionnal Laplacian
- invariance by composition by holomorphic maps
- Homework: exercices 1, 2, 3 (chapter 3)
Lecture 18: March 12th
- Definition of plurisubharmonic functions: usc and the restriction to any line is subharmonic
- Properties directly derived from subharmonicity: stability by decreasing sequences, stability by composition by convex increasing functions, by max and sup
- Local integrability and regularization of psh functions
- Characterization of smooth psh functions in terms of their complex hessians
- Stability of psh functions by composition by holomorphic maps.
- Complement 1: locally integrable functions are psh iff their complex hessian is positive in a distributional sense
- Complement 2: L¹_loc-topology
- Notes for the lectures 17 and 18 (by S. Asgarli)
Lecture 19: March 19th
- pseudo-convex domains: definition
- First characterizations in terms of the existence of a continuous psh exhaustion function
- Geometric characterization in terms of families of disks
- Pseudo-convexity is a local property of the boundary
- slides
Lecture 20: March 24th
- Existence of a smooth psh exhaustion function on any pseudo-convex domain
- A domain with smooth boundary is pseudo-convex iff its Levi form is positive.
- slides
Lecture 21: March 26th
- calculus on (p,q)-forms
- resolution of dbar-equation on pseudo-convex domains
- pseudo-convex domains are domains of holomorphy, and vanishing of Dolbeault cohomology
- Behnke-Thullen's theorem
- slides
Lecture 22: March 31st
- extension of holomorphic functions from subvarieties to pseudo-convex domains
- topology of pseudo-convex domains
- A glimpse at Stein manifolds
- slides
Lecture 23: April 2nd
- Hormander solution to the dbar-equation using L² techniques (I)
Lecture 24: April 7th
- Hormander solution to the dbar-equation using L² techniques (II)
- slides for the last two lectures