# ADVANCED GEOMETRY 1

## Learning outcomes of the course unit

Introduction to complex geometry. Cohomological aspects of complex manifolds.

At the end of the course, students will be familiar with the theory of holomorphic function of several complex variables, with the differential calculus on complex maniflds and with the Hodge theory on compact Hermitian manifolds. They will also be able to tackle the resolution of problems of a theoretical and practical nature in the area of complex differential geometry.

## Course contents summary

Complex Geometry

## Course contents

1. Complex manifolds.

1.1 Introduction to the theory of holomorphic functions of several complex variables.

1.2 Complex structures. Complex projective spaces. Complex tori.

1.3 Almost complex structures. Newlander-Nirenberg theorem.

1.4 (p,q)-forms on complex manifolds. del-bar operator.

1.5 Dolbeault complex.

2. Sheaves and cohomology.

2.1 Pre-sheaves and sheaves.

2.2 Cech cohomolgy.

2.3 Resolutions.

3. Kaehler manifolds.

3.1 Hermitian and Kaehler metrics.

3.2. Kaehler metrics in local coordinates. Examples of Kaehler manifolds.

3.3. Curvature of Kaehler manifolds.

3.4 Cohomological properties of compact Kaehlermanifolds.

3.5 The del-del-bar Lemma.

3.6 Formality of compact Kaehler manifolds.

3.7 Massey products.

4. Introduction to the theory of deformations of complex structures

4.1 Complex analytic families of compact complex manifolds.

4.2 Infinitesimal deformations.

4.3 Differential Graded Algebras.

4.4 del-bar operator and Maurer-Cartan equation.

4.5. Kodaira and Spencer Stability Theorem.

## Recommended readings

J. Morrow, K. Kodaira, Complex manifolds. Reprint of the 1971 edition with

errata. AMS Chelsea Publishing, Providence, RI, 2006. x+194.

D.Huybrechts, Complex Geometry: An Introduction, Springer Universitext, 2014

## Teaching methods

Lectures and classroom exercises.

The lectures will be organized as follows:

in presence with the possibility of enjoying the lessons also remotely in synchronous mode (via Teams). To promote active participation in the course, various individual and small group activities may possibly be proposed,

through the use of the resources present in Elly.

During lectures in traditional mode, the

topics will be formally and rigorously presented. There will be exercises using the elly platform. The course will give particular emphasis to application and computation aspects, while not letting out

the theoretical aspect. The classroom exercises are aimed at showing how and where the abstract results can be applied to make the students understand better the relevance of what they are studying.

## Assessment methods and criteria

The final exam consists of an oral test and the resolution of exercises assigned during the course on the elly platform.

If due to the persistence of the health emergency it was

necessary to integrate with the remote modality the carrying out of the exams will proceed as follows:

remote oral questions.

## Other informations

Exercises and problems to be performed outside of class hours will be assigned by the teacher.

Notes from the teacher will be distributed. The course notes in PDF format and all the material used during the lessons and exercises are made available to the students on the Elly educational platform.