for this workshop
The complex Monge-Ampere equation
at the
American Institute of Mathematics, San Jose, California
organized by
Zbigniew Blocki, Mihai Paun, and Valentino Tosatti
This workshop, sponsored by AIM and the NSF, 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.
The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.
The deadline to apply for support to participate in this workshop has passed.
For more information email workshops@aimath.org
