at the

American Institute of Mathematics, San Jose, California

organized by

Zbigniew Blocki, Mihai Paun, and Valentino Tosatti

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.

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