Informal Complex Geometry
Reading Seminar

Academic Year 2013-14

Organizers: V. Tosatti, B. Weinkove, S. Zelditch

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The seminar met on Fridays from 3.00pm to 4.30pm in Lunt 107.

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Schedule

• October 4
  Yu Wang - The Krylov-Safonov Harnack estimate - Reference

• October 11
  Zhiqin Lu (UC Irvine) - Szegő kernels and Poincaré series - Abstract

  I shall talk about the relations between the Bergman kernels on a compact Kähler manifold and the Bergman kernels on its universal cover. We prove that the summation of the Bergman kernels (over the fundamental group) on the universal cover is equal to the Bergman kernels on the compact manifold, a result that was known only for the case of locally Hermitian symmetric spaces. Using the result, we give a simple proof of Napier's holomorphic convexity theorem and prove the surjectivity of the Poincaré map. This is joint work with S. Zelditch.

• October 18
  Yu Wang - The Krylov-Safonov Harnack estimate - Reference

• October 25
  Xiaokui Yang - The openness conjecture for plurisubharmonic functions - Reference

• November 8 - Special Time and Location: 1:00pm - 2:00pm in Swift 107
  Zbigniew Błocki (Jagiellonian University and Indiana University) - Bergman kernel and pluripotential theory - Abstract

  We discuss some applications of pluripotential theory for the Bergman kernel and metric. In particular, the pluricomplex Green function can be used to prove Bergman completeness of a large class of domains (much bigger that had been known before). It is also applied to estimate the Bergman kernel from below, and recently very accurate estimates of this kind have been obtained. The main tools are the ∂̅-operator and the Monge-Ampère equation. Most of the results are new and nontrivial already in dimension one. For convex domains such a lower bound can be used to simplify Nazarov's complex analytic approach to the Bourgain-Milman inequality in convex analysis.

• November 15
  Tristan Collins (Columbia University) - K-stability for Sasakian manifolds - Abstract

  I will introduce a notion of K-semistability for Sasakian manifolds. This extends to the irregular case the orbifold K-semistability of Ross-Thomas. Our main result is that a Sasakian manifold with constant scalar curvature is necessarily K-semistable. As an application, I will show how one can recover the volume minimization results of Martelli-Sparks-Yau, and the Lichnerowicz obstruction of Gauntlett-Martelli-Sparks-Yau from this point of view. Joint with G. Székelyhidi.

• November 22
  Steve Zelditch - Lower bounds on the Hausdorff measure of nodal sets - References 1 2

• January 10
  Ben Weinkove - Regularity of the Kähler-Ricci flow - Reference

• January 17
  Valentino Tosatti - Finite-time singularities of the Kähler-Ricci flow - Reference

• January 24
  Valentino Tosatti - Finite-time singularities of the Kähler-Ricci flow - Reference

• January 31
  Valentino Tosatti - Finite-time singularities of the Kähler-Ricci flow - Reference

• February 7
  Xiaokui Yang - The strong openness conjecture for plurisubharmonic functions - Reference

• February 14
  Xiaokui Yang - The strong openness conjecture for plurisubharmonic functions - Reference

• February 21
  Albert Chau (UBC) - Estimates for the Kähler-Ricci flow on non-compact Kähler manifolds - Abstract

  I will review some well known a priori estimates in Kähler-Ricci flow, and some modifications of these. Applications to the flow of non-compact Kähler manifolds with unbounded curvature will also be discussed.

• February 28
  Yu Wang - The Evans-Krylov Theorem - Reference

• March 7
  Valentino Tosatti - The volume of a line bundle - Reference

• April 3 - Special Time and Date: 3:00pm - 4:00pm
  Blaine Lawson (Stony Brook) - Analysis on almost complex manifolds - Abstract

  The powerful use of pseudo-holomorphic curves in symplectic topology has kindled interest in the study of almost complex geometry. This talk will discuss several basic analytic results on almost complex manifolds (X, J). They will include: the equivalence of all definitions of J-plurisubharmonicity, the homogeneous and inhomogeneous Dirichlet Problem for the J-Monge-Ampère operator, and a smooth approximation theorem for J-plurisubharmonic functions (recent work with Harvey and Plis)

• April 18
  Ben Weinkove - Global C² estimates for convex solutions of curvature equations - Reference

• April 25
  Valentino Tosatti - A sharp lower bound for the complex singularity exponent of a psh function - Reference

• May 2
  Xiaokui Yang - Positivity of relative adjoint classes - Reference

• May 9
  Valentino Tosatti - Improved Kato inequalities in geometry

• May 16 - Special Time: 3:30pm - 4:30pm
  Ronan Conlon (UQAM) - Some new examples of non-compact Calabi-Yau manifolds - Abstract

  I will present some new examples of non-compact Ricci-flat Kähler manifolds, modelled on a Ricci-flat Kähler cone at infinity. This is joint work with Hans-Joachim Hein (Nantes).

• May 23
  Ben Weinkove - Interior W^2,1+ε estimates for the Monge-Ampère equation - Reference

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Schedule for the past years:

2012-13