Math 262: Geometry of the Complex Monge-Ampère Equation

Monday, Wednesday, Friday, 12-1pm, Science Center 507

Course website: math.harvard.edu/~tcollins/Math262.html

Email: tcollins at math dot harvard dot edu
Office: Science Center 233

(GH) Griffiths-Harris: Principles of Algebraic Geometry
(Sz) Székelyhidi: Introduction to Extremal metrics (PDF)
(PSS) Phong-Song-Sturm: Complex Monge-Ampere Equations (PDF)
(Gut) Gutiérrez: The Monge-Ampère Equation
(W) Weinkove: The J-flow in higher dimensions... (PDF)
(Bl) Blocki: On uniform estimate in the Calabi-Yau theorem (PDF)
(To) Tosatti: Limits of Calabi-Yau metrics when the Kähler class degenerates (PDF)
(Ro) Rong: Convergence and collapsing theorems in Riemannian geometry
(CCT) Cheeger-Colding-Tian: On the singularitites of spaces... (Journal)
(D) Demailly: Complex Analytic and Differential Geometry (PDF)
(Ko) Kolodziej: The complex Monge-Ampere equation and pluripotential theory
(GZ) Guedj-Zeriahi: Intrinsic capacities on compact Kahler manifolds (PDF)
(BT) Bedford-Taylor: The Dirichlet problem for a complex Monge-Ampere equation (Journal)
(EGZ) Eyssidieux-Guedj-Zeriahi: A priori L^∞-estimates for degenerate complex Monge-Ampere equations (PDF)
(DP) Demailly-Paun: Numerical Characterization of the Kahler cone (PDF)
(CoT) Collins-Tosatti: Kahler currents and null loci (PDF)
(RZ) Rong-Zhang: Continuity of extremal transitions and flops for Calabi-Yau manifolds (PDF)
(RZ2) Rong-Zhang: Degenerations of Ricci-flat Calabi-Yau manifolds (PDF)
(To2) Tosatti: Adiabatic limits of Ricci-flat Kahler metrics (PDF)

This is approximate. I will attempt to keep track of all the references used in class.

Date	Material	Problems/Handouts
Jan. 26-30	Kahler Manifolds: References: (GH)-Chapter 0, (Sz)-Chapter 1
Feb. 2-6	Yau's Theorem The C2 and C3 estimates for the Complex Monge-Ampère equation: (PSS), (Sz)- Chapter 3. The Pogorelov estimate for the real Monge-Ampère equation: (Gut)- Chapter 4.
Feb. 9-13	Yau's Theorem The C0 estimate. The Moser iteration (PSS). The exponential Moser iteration (W).
Feb. 16-20	Yau's Theorem, Limits of CY manifolds The ABP estimate (Bl), (PSS). Uniform Diameter Bound (To).
Feb. 23-27	Limits of CY manifolds The Diameter Bound (To).
Mar. 2-6	Limits of CY manifolds Gromov-Hausdorff Convergence, Gromov's compactness theorem (Ro),(CCT)
Mar. 9-13	Pluripotential Theory Currents, PSH functions (D), (Ko), (GH)
Mar. 23-27	Pluripotential Theory The Bedford-Taylor theory of the Monge-Ampere operator (BT), (D), (GZ) and the C0 estimate of Kolodziej (PSS), (Ko), (EGZ)
Mar. 30 - Apr. 3	Pluripotential Theory and the Non-collapsing case Existence of Kahler currents in nef classes (DP) and the non-Kahler locus (CoT)
Apr. 6-10	The Non-Collapsing Case Uniform estimates, identification of the GH limit (To), (CoT), (RZ), (RZ2)
April 13-15	The Collapsing Case Uniform Estimates (To2)