**1 feb 2016 - 31 may 2016**

Dipartimento di Matematica e Informatica "U. Dini", Università di Firenze

**Title:** "Complex Monge-Ampère equations on compact Kähler manifolds"

**Number of hours:** 30h / 6 CFU

(20h by Chinh, 10h of preparation in complex geometry by Daniele)

**Period:** starting from the middle of February until middle of May

**Place:** Firenze

**Program:**

The aim of the course is to survey recent important results concerning the complex Monge-Ampère equation on compact Kähler manifolds. We first present the proof of the Calabi conjecture by reducing it to a complex Monge-Ampère equation and establishing a priori estimates. We
then go on to develop a (global) pluripotential theory on compact Kähler manifold, focusing on the theory of (generalized) capacities and its application in proving singular uniform estimate. We also introduce a variational method to solve degenerate complex Monge-Ampère equation and study the finite energy spaces which will be recognized as the completion of the space of Kähler metrics equipped with the Mabuchi distance. The last lecture will be a more advanced talk (materials from previous lectures will be useful) on the weak Calabi flow, its large time behavior as well as some conjectures in
the field.

A preliminary part of the course will discuss basic results on complex and Kähler geometry.

**Teacher:** Chinh H. Lu

Centro di Ricerca Matematica "Ennio de Giorgi", Pisa

chinh.lu (AT) sns.it

**Collaborator:** Daniele Angella

DiMai "Dini", Università di Firenze

daniele.angella (AT) gmail.com, daniele.angella (AT) unifi.it

**Organizers:**
Daniele Angella,
Simone Calamai.

**Courses:**

1 Feb 2016: **:** *preliminary meeting* Aula 7, DiMaI Dini, Firenze

8 Feb 2016: **D. Angella:** *Preliminary lectures on Complex Geometry, 1* Aula 7, DiMaI Dini, Firenze

15 Feb 2016: **D. Angella:** *Preliminary lectures on Complex Geometry, 2* Aula 7, DiMaI Dini, Firenze

22 Feb 2016: **D. Angella:** *Preliminary lectures on Complex Geometry, 3* Aula 7, DiMaI Dini, Firenze

4 Mar 2016: **D. Angella:** *Preliminary lectures on Complex Geometry, 4* Aula 1, DiMaI Dini, Firenze

7 Mar 2016: **D. Angella:** *Preliminary lectures on Complex Geometry, 5* Aula 7, DiMaI Dini, Firenze

9 Mar 2016: **C. H. Lu:** *Complex Monge-Ampère equations on compact Kähler manifolds, 1* Aula Tricerri, DiMaI, Firenze

18 Mar 2016: **C. H. Lu:** *Complex Monge-Ampère equations on compact Kähler manifolds, 2* Aula Tricerri, DiMaI, Firenze

1 Apr 2016: **C. H. Lu:** *Complex Monge-Ampère equations on compact Kähler manifolds, 3* Aula Tricerri, DiMaI, Firenze

6 Apr 2016: **C. H. Lu:** *Complex Monge-Ampère equations on compact Kähler manifolds, 4* Aula Tricerri, DiMaI, Firenze

12 Apr 2016: **C. H. Lu:** *Complex Monge-Ampère equations on compact Kähler manifolds, 5* Aula Tricerri, DiMaI, Firenze

15 Apr 2016: **C. H. Lu:** *Complex Monge-Ampère equations on compact Kähler manifolds, 6* Aula Tricerri, DiMaI, Firenze

2 May 2016: **C. H. Lu:** *Complex Monge-Ampère equations on compact Kähler manifolds, 7* Aula Tricerri, DiMaI, Firenze

3 May 2016: **C. H. Lu:** *Complex Monge-Ampère equations on compact Kähler manifolds, 8* Aula Tricerri, DiMaI, Firenze

**Seminars:**
** 6 Jun 2016**

10:00 -
S. Lohove:
A characterization of the Class \mathcal{E}(X,\omega)

11:00 -
D. Berti:
Uniqueness of solutions to Monge-Ampère equations

11:40 -
N. Lombardi:
On C^0-estimates

**Speakers:**
Chinh Hoang Lu.