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
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):
Possible other topics:
Now here's what actually happened:
If only I'd had more time, I would've
- 1/19-1/28: Subharmonic functions and
- 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
- 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.
As it is, that's all folks. Thanks for your attention.
- 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).