Universität Leipzig (Leipzig) Felix-Klein-Hörsaal

# Chow Lectures 2017

*Wei-Liang Chow* (October 1, 1911, Shanghai – August 10, 1995, Baltimore) was a celebrated mathematician best known for his work in algebraic geometry, though he also made important contributions to other areas such as differential geometry, differential equations, and control theory. Within algebraic geometry he is known among other things for his work on intersection theory (Chow's moving lemma, Chow ring), for Chow's theorem on algebraicity of projective analytic varieties, and for many other achievements.

Chow was born in Shanghai, had his school education in the United States, and graduated from the University of Chicago in 1931. He obtained his PhD in **Leipzig** in 1936, where he worked with van der Waerden (who was a professor at the University of Leipzig), introducing in particular Chow coordinates. Subsequently, Chow was a professor in Nanjing, Princeton and at Johns Hopkins University were he worked until 1977.

The idea behind the Chow lectures - which almost coincide with Chow's 80th Phd anniversary - is to give students the opportunity to enjoy lectures by internationally renowned experts on active fields of modern mathematics and to create a stimulating research environment.

Inaugural speaker for the Chow lectures 2017 is *Claire Voisin* (Collège de France). Claire Voisin is mainly known for her works on algebraic geometry and its relation to Hodge theory. Among her fundamental achievements are counterexamples to the Kodaira problem and a solution of the generic case of Green's conjecture. She obtained several distinctions such as the European Mathematical Society Prize in 1992, the Servant Prize in 1996, the Sophie Germain Prize in 2003 and the Clay Research Award in 2008. In 2016 she received the Gold medal from the French National Centre for Scientific Research – the highest scientific research award in France.

In addition, there will be a historical introduction by Norbert Schappacher (Université de Strasbourg).

## Plan of preparatory sessions (subject to change)

### Monday

- Varieties vs. Manifolds
- Definition of complex manifold and Kähler manifold
- Smooth projective complex varieties are compact Kähler manifolds, Chow's Theorem
- Running example: projective n-space over C

- Hodge theory
- Definition of Hodge structure, Hodge classes and Hodge numbers
- Hodge decomposition theorem
- (real and complex) De Rham cohomology, Comparison Theorem
- Idea for Hodge decomposition: Direct sum decomposition of differential forms with complex coefficients

### Tuesday

- Motivation: Cycle class map
- Definition of Chow groups and Chow rings
- How to compute Chow groups
- Pushforward, pullback, localisation sequence
- Examples: Affine and projective space
- Further examples (if time allows): correspondences, Chow groups of projective bundles and blow ups

See this file with the easy exercises. The participants of the Preparatory Lectures should solve the exercises BEFORE they are coming to the first lecture.

We want to stress that the Exercises are meant to teach basic definitions and concepts. They are all easy and should not require too much time, but playing around with the objects we will be dealing with will significantly improve the level of understanding when visiting the Chow and the Preparatory Lectures.

We will NOT repeat the notions introduced in the Exercises during our Preparatory Lectures, but assume that they are known. Of course, participants are welcome and encouraged to ask questions about the Exercises before, during and after our Preparatory Lectures.

The organisers of the preparatory lectures are: