JHU-UMD Complex Geometry Seminar Archives for Academic Year 2014


The Higgs bundle moduli space and L^2 cohomology

When: Tue, September 10, 2013 - 4:30pm
Where: MTH 3206, Colloquium
Speaker: Jan Swoboda (Bonn & Stanford) -
Abstract: I will review some known results and several open questions concerning the L^2 cohomology of various non-compact moduli spaces carrying a hyperkähler metric. Particular attention will be given to the moduli space M of Higgs bundles over a Riemann surface, which arises as the space of solutions of a certain system of nonlinear PDEs, called self-duality equations. Although many aspects of M as a hyperkähler manifold are by now well-established, it is still an open problem to describe the geometry of this non-compact manifold near its ends. In this talk, I shall present some preliminary results in that direction. (Joint work with Rafe Mazzeo, Hartmut WeiÃ, and Frederik Witt).

The boundary of the Kahler cone

When: Tue, September 24, 2013 - 4:30pm
Where: Krieger Hall 300
Speaker: Tristan Collins (Columbia University) - http://www.math.columbia.edu/~tcollins/homepage.html
Abstract: We will discuss a geometric characterization of classes of positive volume on the
boundary of the Kahler cone of a compact Kahler manifold. As an application,
we will show that finite time singularities of the Kahler-Ricci flow always form along analytic subvarieties. This work is joint with Valentino Tosatti.

On critical exponents for some complex Monge-Ampere equations

When: Tue, October 8, 2013 - 4:30pm
Where: Krieger Hall 300, JHU
Speaker: Chi Li (Stony Brook)
http://www.math.sunysb.edu/~chili/
Abstract: I will talk about how to determine the critical exponents of
some complex Monge-Ampere equations arising from Kahler-Einstein problem,
both in the global and local settings. Blow-up behaviors will also be
discussed.

Some Results in Complex Hyperbolic Geometry

When: Tue, October 22, 2013 - 4:30pm
Where: MATH 3206
Speaker: Luca F. Di Cerbo (Duke)
http://fds.duke.edu/db/aas/math/luca

Abstract: In this talk, I present a new approach to the study of cusped complex
hyperbolic manifolds through their compactifications. Among other things, I
give effective bounds on the number of complex hyperbolic manifolds with given
upper bound on the volume. Moreover, I estimate the number of cuspidal ends of
such manifolds in terms of their volume. Finally, I address the classification
problem for cusped complex hyperbolic surfaces with minimal volume. This is the
noncompact or logarithmic analogue of the well known classification problem for
fake projective planes.


Asymptotically Conical Calabi-Yau manifolds

When: Tue, November 12, 2013 - 4:30pm
Where: Math 3206
Speaker: Ronan J. Conlon
Abstract: Asymptotically Conical (AC) Calabi-Yau manifolds are Ricci-flat
Kahler manifolds that resemble a Ricci-flat Kahler cone at infinity. The
first result I will present is a refinement of an existence theorem of
Tian and Yau from the early '90's for these manifolds.

In more recent years, new examples of "irregular" Calabi-Yau cones have
been discovered by mathematicians and physicists alike. I will also
present the first example of an AC Calabi-Yau metric on a smoothing of
such a Calabi-Yau cone.

This is joint work with Hans-Joachim Hein (Nantes).

Degeneration of Kahler-Ricci Solitons

When: Tue, November 26, 2013 - 4:30pm
Where: JHU Krieger Hall 300
speaker: Jacob Sturm

Abstract: We shall discuss the proof of partial C^0 estimates for Kahler-Ricci solitons with bounded Futaki invariant, generalizing the recent work of Donaldson-Sun in the setting of Fano Kahler-Einstein manifolds. In particular, any sequence of bounded solitons has a convergent subsequence in the Gromov-Hausdorff topology to a Kahler-Ricci soliton on a Q-Fano variety with log terminal singularities. This is joint work with D.H. Phong and Jian Song.

Bergman kernel and pluripotential theory

When: Thu, December 12, 2013 - 3:30pm
Where: B0425 (basement)
Speaker: Zbigniew Blocki (Jagiellonian)
http://gamma.im.uj.edu.pl/~blocki/

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 dbar-operator
and the Monge-Ampere 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.

Singular Kahler-Einstein metrics of small angles

When: Wed, January 1, 2014 - 7:00am
Where: Math 3206
Speaker: Jesus Martinez-Garcia (JHU) -
Abstract: The existence of a Kahler-Einstein metric on a Fano variety is equivalent to the algebro-geometric concept of K-stability. However K-stability is very difficult to test. For those Fano varieties which are not K-stable, we can define a singular Kahler-Einstein metric known as Kahler-Einstein metric with edge singularities, depending on a parameter \beta\in (0,1]. These metrics also have a reformulation in terms of log K-stability. It is well known that a smooth del Pezzo surface admits a Kahler-Einstein metric if and only if it is not the blow-up of P^2 in one or two points. However they always admit a Kahler-Einstein edge metric. In this talk, after introducing all these topics, I explain how we can use birational geometry and log canonical thresholds to find Kahler-Einstein edge metrics on all del Pezzo surfaces. This is joint work with Ivan Cheltsov.

TBA

When: Tue, February 11, 2014 - 4:30pm
Where: JHU Krieger Hall 300
Speaker: Dror Varolin ((Stony Brook) ) -

Beyond the space of Kahler metrics

When: Tue, February 18, 2014 - 4:30pm
Where: UMD Math 3206
Speaker: Darvas Tamas (Purdue)

Abstract: Given a compact Kahler
manifold, according to Mabuchi, the set of Kahler forms in a fixed
cohomology class has the natural structure of an in finite dimensional
Riemannian manifold. This space is not geodesically convex, as Kahler
forms along a geodesic segment connecting two arbitrary points are
typically not smooth. We address this issue by extending the Mabuchi
metric structure to Kahler forms with less regularity. In the resulting
metric space every two points can be joined by an honest geodesic, what is
more, this space will be non-positively curved in the sense of Alexandrov.

Szego kernels and Poincare series

When: Tue, March 4, 2014 - 4:30pm
Where: JHU Ames 218
Speaker: Zhiqin Lu
http://www.math.uci.edu/~zlu/

Abstract: Let (M,L) be a polarized Kaehler manifold. We prove the
folklore theorem that the Bergman kernel of M is equal to the Poincare
series of the L^2 Bergman kernel on its universal cover. We apply this
result to give a simple proof of a theorem of Napier and the surjectivity
of the Poincare series. On a more general setting, I shall also discuss
the Agmon-type estimate when the injectivity radius of the manifold is
sufficiently small. This is the joint work with Steven Zelditch.

Homological Mirror Symmetry and Geometry

When: Tue, April 8, 2014 - 3:00pm
Where: Ames 218
Speaker: Ludmil Katzarkov, Univ. of Miami

Abstract:
Homological Mirror symmetry is a categorical correspondence originating
from string theory. In mathematics it is known as a way of counting holomorphic
curves. We will explain some less known connections of Homological
Mirror Symmetry with birational geometry and theory of algebraic cycles.



Nodal sets of eigenfunctions

When: Tue, April 22, 2014 - 4:30pm
Where: Ames 218
Speaker: Steve Zelditch, Northwestern University,
http://mathnt.mat.jhu.edu/zelditch/

Abstract: Nodal (zero) sets of eigenfunctions are important in physics
and are also analogues of real algebraic varieties on Riemannian
manifolds. This analogy is close if the metric is real analytic, and I
will present results on the complex geometry of nodal sets. When the
metric is only smooth, much less is known. I will present recent results
with J. Jung proving that the number of nodal domains tends to infinity
with the eigenvalue if the geodesic flow is ergodic.