Informal Complex Geometry
Reading Seminar
Academic Year 201314
Organizers: V. Tosatti, B. Weinkove, S. Zelditch
The seminar met on Fridays from 3.00pm to 4.30pm in
Lunt 107.
Schedule

October 4
Yu Wang  The KrylovSafonov 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 KrylovSafonov 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 MongeAmpère 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
BourgainMilman inequality in convex analysis.

November 15
Tristan Collins (Columbia University)  Kstability for Sasakian manifolds  Abstract
I will introduce a notion of Ksemistability for Sasakian manifolds. This extends to the irregular case the orbifold Ksemistability of RossThomas. Our main result is that a Sasakian manifold with constant scalar curvature is necessarily Ksemistable. As an application, I will show how one can recover the volume minimization results of MartelliSparksYau, and the Lichnerowicz obstruction of GauntlettMartelliSparksYau 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 KählerRicci flow  Reference

January 17
Valentino Tosatti  Finitetime singularities of the KählerRicci flow  Reference

January 24
Valentino Tosatti  Finitetime singularities of the KählerRicci flow  Reference

January 31
Valentino Tosatti  Finitetime singularities of the KählerRicci 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 KählerRicci flow on noncompact Kähler manifolds  Abstract
I will review some well known a priori estimates in KählerRicci flow, and some modifications of these. Applications to the flow of noncompact Kähler manifolds with unbounded curvature will also be discussed.

February 28
Yu Wang  The EvansKrylov 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 pseudoholomorphic 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
Jplurisubharmonicity, the homogeneous and inhomogeneous
Dirichlet Problem for the JMongeAmpère operator,
and a smooth approximation theorem for Jplurisubharmonic
functions (recent work with Harvey and Plis)

April 18
Ben Weinkove  Global C^{2} 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 noncompact CalabiYau manifolds  Abstract
I will present some new examples of noncompact Ricciflat
Kähler manifolds, modelled on a Ricciflat Kähler cone at infinity. This
is joint work with HansJoachim Hein (Nantes).

May 23
Ben Weinkove  Interior W^{2,1+ε} estimates for the MongeAmpère equation  Reference
Schedule for the past years:
201213