Math 80380 (Topics in Complex Analysis)

The Complex Monge-Ampere Operator

Instructor: Jeffrey Diller (click for contact info)
Time and place: MWF, 10:40-11:30 in 131 DeBartolo.

Abstract: The real and imaginary parts of a holomorphic function of one complex variable are harmonic. So begins a beautiful relationship between classical complex function theory and the (linear) Laplace equation in the plane. In several complex variables, however, the Laplace operator and its associated potential theory are not so relevant. Instead, one is led to consider the complex Monge-Ampere operator. This course aims to tell the much more recent story of this non-linear operator and its applications to complex analysis and geometry.

Text: I'm putting the books Pluripotential Theory by Klimek and The Complex Monge-Ampere Operator and Pluripotential Theory by Kolodziej on reserve in the math library. The former is more elementary but also a little dated by now. Mostly, however, I'll be following freely available notes and articles which I'll link to below. In particular, for the first half of the semester, I'll lean particularly heavily on some course notes by Blocki. And now I find I've been typing up some notes of my own, partly to clean up issues from lecture and partly just to make myself happy. These aren't comprehensive, but they do include some extra detail and different presentation of many basic points.

Topics covered (the plan):

A review of the Laplace operator and distributions (Klimek's book, notes by Blocki)
Plurisubharmonic functions and positive closed currents (Klimek, Blocki, Kolodziej)
The complex Monge-Ampere (CMA) operator (Klimek, Blocki, Kolodziej)
The Dirichlet problem for CMA on strictly pseudoconvex domains (Blocki, Kolodziej)
Compact Kahler manifolds (Kolodziej, Blocki notes on the Calabi conjecture)
Application to holomorphic dynamics (Sibony notes)
Variational approach to the inhomogeneous CMA equation on a compact Kahler manifold (articles by Boucksom-Berman and Boucksom-Berman-Guedj-Zeriahi)
Kahler-Einstein metrics (BBGZ article, Blocki notes on Calabi conjecture)

Possible other topics:

Polynomial approximation and transfinite diameter (Boucksom-Berman article and other article, Levenberg notes)
Lelong numbers and Intersection theory (Demailly survey)
More complex dynamics (Sibony notes)

Now here's what actually happened:

1/19-1/28: Subharmonic functions and distributions
1/31-2/4: Instructor flees to Marseille for conference; vague promises about making up missed lectures
2/7-2/9: Plurisubharmonic functions
2/11-2/18: (positive) forms and currents in C^n
2/21: Instructor's trip to Banff falls through, but class still canceled (total of 3 classes to make-up now).
2/23-3/9: Definition and basic properties of complex Monge-Ampere: CLN estimate, continuity under monotone limits, capacity and quasicontinuity, comparison principle.
3/11-4/6: the Dirichlet problem for CMA in C^n.
4/8-4/13: Positive closed (1,1) on projective space and homogeneous potentials.
4/15--4/27: Holomorphic dynamics on projective space.
4/29--5/2: Complex Monge-Ampere on compact Kahler manifolds: the case of Riemann surfaces
5/4--5/10: CMA on compact Kahler mflds: the general case.
OK, I ran out of time and wasn't able to prove that functions maximizing E_\mu necessarily solve CMA. The missing point is a differentiability result due to Berman and Boucksom. Here are some notes to fill the gap.

If only I'd had more time, I would've

Proven uniqueness for the variational solution of CMA (1 lecture)
Shown how to modify the variational approach to produce Kahler-Einstein metrics (2 lectures).
Discussed work of Berman and Boucksom on pluripotential theory and polynomial approximation (N >> 0 lectures, but a key feature is their differentiability result again).

As it is, that's all folks. Thanks for your attention.