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:

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).

**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.