The course is supported by the project SIR2014 "Analytic aspects in complex and hypercomplex geometry" and it is part of the PhD program in Mathematics of Università di Firenze.


Best regards,

Daniele Angella, Simone Calamai






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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 metircs 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@sns.it
  https://sites.google.com/a/sns.it/chinh-h-lu/home

Collaborator: Daniele Angella
  DiMai "Dini", Università di Firenze
  daniele.angella@gmail.com, daniele.angella@unifi.it
  https://sites.google.com/site/danieleangella/