Complex Geometry


Lecturer

Enric Solé-Farré (Chern Instructor, BIMSA)
Date

2025-09-23 ~ 2025-12-18
Schedule
Weekday    Time    Venue    Online    ID    Password
Tue,Thu    10:40-12:15    A14-202    Zoom    293 812 9202    BIMSA

Prerequisite

Complex analysis and differential geometry. Familiarity with Riemannian geometry and vector bundles is desirable.
Introduction

This graduate-level course offers an introduction to the fundamental concepts and techniques of complex differential geometry.

The central aim of the course is to understand the criteria that determine when a compact complex manifold can be realized as a smooth projective algebraic variety. This is the celebrated Kodaira Embedding Theorem, a cornerstone result that provides a precise differential-geometric condition (the existence of a positive line bundle, or a Hodge metric) for a complex manifold to be projective (and thus algebraic by Chow's theorem). We will work through the necessary machinery to fully prove this theorem.

Time permitting, we will then discuss Kodaira-Spencer deformation theory and discuss the case of Calabi-Yau manifolds, studied by Tian-Todorov.
Syllabus

0)Overview
1) Holomorphic functions
2) Complex and almost complex manifolds
3) Vector bundles and sheaves
4) Kodaira dimension and Siegel's theorem
5) Divisors and blow-ups
6) Metrics and connections
7) The K?hler condition
8) Positivity and vanishing
9) The Kodaira embedding theorem
10) Kodaira-Spencer deformation theory
11) (Formal) Tian-Todorov theorem
Reference

D. Huybrechts, Complex Geometry: An Introduction
J.-P. Demailly, Complex Analytic and Differential Geometry
Video Public

No
Notes Public

Yes
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FROM 202.120.11.*