# The complex Monge-Ampere equation

August 15 to August 19, 2016

at the

American Institute of Mathematics, San Jose, California

organized by

Zbigniew Blocki, Mihai Paun, and Valentino Tosatti

## Original Announcement

This workshop will be devoted to the complex Monge-Ampere equation and its applications in complex geometry and analysis.

The main topics of the workshop are:

• The Iitaka conjecture: the approach outlined by H. Tsuji for this famous problem in complex algebraic geometry makes use of a family of complex Monge-Ampere equations with degenerate right hand side, and the crux of the matter is understanding the variation of solutions of such equations.

• The partial $C^0$ estimate: this result states the existence of a uniform lower bound for the Bergman kernel associated to a power of an ample Hermitian line bundle, provided that we fix a few natural geometric invariants, and was recently proved by Donaldson and Sun. As it stands by now, the proof of this result is using in an essential manner subtle differential-geometric techniques, namely the Gromov-Hausdorff convergence theory. On the other hand, it can also be viewed as a quantitative version of a Fujita-type theorem in algebraic geometry. We intend to analyze further the analogy between these important achievements in complex geometry.

• Symmetrization and complex isoperimetric inequalities: the main part of a famous result of Kolodziej is a local $L^p$-estimate for the complex Monge-Ampere equation. The proof is very technical, and uses pluripotential theory. It would interesting to obtain a purely PDE proof of this result. The work of Talenti, Tso and Trudinger on real operators suggests that the right approach should be through a symmetrization result for the complex Monge-Ampere equation, which would follow from conjectural complex isoperimetric inequalities.

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

A list of open problems.

Papers arising from the workshop:

On the $C^1,1$ regularity of geodesics in the space of Kähler metrics
by  Jianchun Chu, Valentino Tosatti, and Ben Weinkove,  Ann. PDE 3 (2017), no. 2, Art. 15, 12 pp.  MR3695402
On the maximal rank problem for the complex homogeneous Monge-Ampère equation
by  Julius Ross and David Witt Nyström