Math 514, Complex Algebraic Geometry (Fall 2020)

### Instructor: Pierre Albin

### Email: palbin [at] illinois .edu

### Logistics, or how will the course run:
The class will be carried out entirely online.
I will provide lecture notes and videos of the lectures.
I will let you know when I plan to record a lecture and you are
invited to join me via Zoom and ask questions in real time.
You are not be required to join the lecture but I hope that if you
can, you will.
Each lecture will have a short quiz for you to take in
Moodle within
48 hours of the lecture.
I will assign homework regularly which you will turn in using
Moodle.
There is also a Piazza
page where I encourage you to post and answer
questions. I will check Piazza regularly.

### Grading:

Quizzes (20%)

Homeworks (80%)

We will drop the two lowest homework scores and the two lowest quiz scores.

### Holidays:
Classes begin on August 24 and end on December 9.
There will be no classes on:

**Labor Day**, September 7,

**Election Day**, November 3,

**Fall Break**, November 21-29

### Description:

This course is an introduction to the geometry of Kähler manifolds
and the Hodge structure of cohomology.

Kähler manifolds are at the intersection of complex analytic
geometry, Riemannian geometry, and symplectic geometry.
Moreover, every smooth projective variety is a Kähler manifold.
All of this structure is reflected in a rich theory of geometric and
topological invariants. In this course we will develop techniques
from sheaf theory and linear elliptic theory to study the cohomology
of Kähler manifolds.

In the setting of complex projective varities one can use
geometric and analytic methods to address questions in algebraic geometry. The
results and examples then serve as guides in more algebraic
approaches.

There is a lot of current research taking place in the setting of
Kähler manifolds. For instance, very recently a couple of teams of
researchers have managed to relate a notion of stability to the existence of Käher-Einstein
metrics on Fano spaces. For another example, one of the Clay millenium
problems is to prove the Hodge conjecture: every Hodge class of a
non-singular projective variety over ** C ** is a rational linear
combination of cohomology classes of algebraic cycles.
This course should be useful for students interested in research in
geometry, be it differential or algebraic, and topology.