FALL 2016: MATH 868D “Pluripotential Theory”

Syllabus: Modern pluripotential theory has its roots in the work of Bedford and Taylor in the 70's. Since then it has become an indispensable tool in many fields, and its use led to many breakthroughs in complex dynamics, differential geometry, complex algebraic geometry. In the beginning weeks I will give a gentle introduction to classical pluripotential theory on C^n, and then using the Perron method construct solutions to the Dirichlet problem for the complex Monge-Ampere equation on a strictly pseudoconvex domain. The second part of the course would be about proving apropri estimates for global complex Monge-Ampere equations, culminating in the proof of the Calabi-Yau theorem, one of the cornerstones of complex differential geometry. The third part would be about the finite energy pluripotential theory of Guedj-Zeriahi and the variational approach to global equations of complex Monge-Ampere type, perhaps touching on recent work resolving Tian's properness conjecture, if there is enough time.

Beginning graduate students with some experience in complex analysis or PDE's (not necessarily both) should be able to follow the course. Occasionally homework will be posted and it is recommended that students do these to get a better grasp of the material. If you are taking this course for credit you will be required to type up some of the lectures.

References:

• [R1] Z. Blocki, The complex Monge-Ampere operator in pluripotential theory


• [R2] Z. Blocki, The Calabi Yau theorem


• [R3] Z. Blocki, The complex Monge-Ampere equation in Kahler geometry


• [R4] V. Guedj, A. Zeriahi, The weigthed Monge-Ampere energy of quasiplurisubharmonic functions


• [R5] V. Guedj, Degenerate complex Monge-Ampere equations and singular Kahler-Einstein metrics


• [R6] S. Bouckson, R. Berman, P. Eyssideux, V. Guedj, A. Zeriahi, Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties


• [R7] T. Darvas, The Mabuchi Completion of the Space of Kähler Potentials


• [R8], T. Darvas, The Mabuchi Geometry of Finite Energy Classes


• [R9] T. Darvas, Y. Rubinstein, Tian's properness conjectures and Finsler geometry of the space of Kahler metrics

Other references will be provided during the course.