Complex analytic varieities

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