November 16, 2016 - March 17, 2017
The aim of the course is to give an introduction to complex analytic
varieties, i.e. sets that are locally the zero sets of holomorphic
functions.
Analytic varieties give rise to a rich and interesting theory and
they appear naturally in many areas of mathematics.
We will start by studying local aspects of the theory, including the
Weierstrass preparation theorem and the local parametrization
theorem. Then we will introduce coherent sheaves, which is a fundamental tool
in the study of analytic varieties
that allows us to glue local data. After investigating various global aspects, we will focus on
singularities of varieties; we will study normal spaces, which have mild singularities in a
certain sense, and discuss blowups and resolution of singularities.
The course is based on Chapter II in Complex Analytic and Differential
Geometry by Jean-Pierre Demailly.
Some suggestions for further reading:
Several Complex Variables with Connections to Algebraic Geometry and
Lie Groups by Joseph L. Taylor
An Introduction to Complex Analysis in Several Variables by
Lars Hörmander
Analytic Functions of Several Complex Variables by Robert
C. Gunning and Hugo Rossi
Local Analytic Geometry by Theo de Jong and Gerhard
Pfister
Commutative Algebra by Oscar Zariski and Pierre Samuel
Differential Forms on Singular Varieties - De Rham and Hodge
Theory Simplified by Vincenzo Ancona and Bernard Gaveau
The course will run through period 2 and 3 2016/2017. In November -
December 2016 we will meet on Mondays 13.15-15.00 in MVL14.
In January-March 2017, starting January 20th, we will meet on Fridays
13.15-15.00 in MVL15, except on January 20th when we will meet in
MVH11. Also there will be no lecture on February 3rd, but
instead a lecture on Monday January 30th, 10.00-11.45 (in MVL15).
Here are the homework problems (this document will be updated throughout
the course).
Preliminary schedule
- November 7, 2016: Introduction, the ring of germs of
holomorphic functions, Weierstrass preparation theorem
- November 14, 2016: Weierstrass division theorem, algebraic
properties of the ring
of germs of holomorphic functions
- November 21, 2016: (Germs of) complex analytic sets,
Local parametrization theorem
- November 28, 2016: Local parametrization theorem, continued
- December 5, 2016: Rückert's Nullstellensatz, regular and singular points,
dimension, (coherent) sheaves
- November 12, 2016: Coherent sheaves, Oka's theorem
- November 19, 2016: Oka's theorem continuted, ideal
sheaves, strong Noetherian property
- January 20, 2017: Repetition, coherence of ideal
sheaves
- January 27, 2017: Characterizations of the singular locus
of an analytic set, complex spaces
- January 30, 2017: Complex spaces, irreducible components,
holomorphic functions on complex spaces
- February 10, 2017: Coherent sheaves on complex spaces,
Riemann Extension Theorem, zeros of holomorphic functions
- February 17, 2017: Divisors, cycles, meromorphic
functions, weakly holomorphic functions
- February 24, 2017: Weakly holomorphic functions, normal
spaces
- March 3, 2017: Normal
spaces, normalization
- March 10, 2017: Resolution of
singularities, blowups, complex analytic schemes.