JHU-UMD Complex Geometry Seminar Archives for Academic Year 2014

Geometry of the space of positive Lagrangians

When: Tue, September 9, 2014 - 3:00pm
Where: MTH 2300
Speaker: Jake Solomon (Hebrew University) -
Abstract: A Lagrangian submanifold of a Calabi-Yau manifold is called positive if the real part of the holomorphic volume form restricted to it is positive. A Hamiltonian isotopy class of positive Lagrangian submanifolds admits a Riemannian metric with non-positive curvature. Its universal cover admits a functional, with critical points special Lagrangians, that is strictly convex with respect to the metric. Solutions of the geodesic equation, both smooth (with A. Yuval) and viscosity (with Y. Rubinstein), will be discussed. Mirror symmetry relates these phenomena with analogous phenomena for the space of Hermitian metrics on a holomorphic vector bundle and the space of Kahler metrics.

Singular Kahler-Einstein metrics of small angles

When: Tue, October 7, 2014 - 4:30pm
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.

Kahler-Einstein metrics on singular Fano varieties

When: Tue, October 21, 2014 - 4:30pm
Where: JHU, Krieger Hall, Room 300
Speaker: Chengjian Yao (SUNY, Stony Brook) -
Abstract: The existence of smooth Kahler-Einstein metrics is equivalent to being polystable for smooth Fano manifolds. The generalization of this equivalence to the situation of singular Fano varieties is an interesting question since it might be used to construct compactified moduli space for K-polystable/Kahler-Einstein Fano manifolds as conjectured by Odaka-Spotti-Sun. In this talk, I will discuss some recent progress about the existence of Kahler-Einstein metrics on singular Fano varieties which is smoothable and K-polystable. This is a joint work with Cristiano Spotti and Song Sun.

Singular solutions to the Monge-Ampere equation

When: Tue, October 28, 2014 - 4:30pm
Where: Math 3206
Speaker: Connor Mooney (Columbia University) -
Abstract: Strictly convex solutions to the Monge-Ampere equation \det D^2u = 1 are smooth. However, there are examples of singular solutions, due to Pogorelov and Caffarelli, that degenerate along line segments. I will discuss recent optimal estimates for the Hausdorff dimension of the singular set and applications to the regularity theory for singular solutions.

The Kahler-Ricci flow and its singularities

When: Tue, November 11, 2014 - 4:30pm
Where: JHU Gilman Hall 219
Speaker: Valentino Tosatti (Northwestern University) -
Abstract: I will give an introduction to the study of Ricci flow on compact Kahler manifolds, and explain how its behavior reflects the structure of the complex manifold. I will then describe a result (joint with T.Collins) which gives a geometric description of the set where finite-time singularities occur, answering a conjecture of Feldman-Ilmanen-Knopf and Campana.

(MADGUYS) Metro Area Differential Geometry Seminar

When: Fri, November 14, 2014 - 10:00am
Where: Colloquim, Room 3206
Speaker: Various Speakers () - http://math.jhu.edu/~bernstein/MDGS/index.html
Abstract: Baltimore-Washington -
(MADGUYS) Metro Area Differential Geometry Seminar

Kaehler-Einstein metrics and higher alpha-invariants

When: Tue, December 2, 2014 - 4:30pm
Where: MTH 3206
Speaker: Heather Macbeth () -
Abstract: In the most delicate cases, the proof of existence of Kaehler-Einstein metrics on Fano surfaces M uses the "alpha-invariants" \alpha_{m,1}(M) and \alpha_{m,2}(M). I will give a survey of that proof (from 1990 and due to Tian), then discuss recent work on what can be learned in higher dimensions from the higher alpha-invariants \alpha_{m,k}(M).

The Yang-Mills flow on Kaehler manifolds

When: Tue, December 9, 2014 - 4:30pm
Where: MTH 3206
Speaker: Richard Wentworth (UMD) -
Abstract: The fundamental work of Donaldson and Uhlenbeck-Yau proves the the smooth convergence of the Yang-Mills flow of stable integrable unitary connections on hermitian vector bundles over Kaehler manifolds. This was generalized by Bando and Siu to incorporate certain (singular) hermitian structures on reflexive sheaves. Bando-Siu also conjectured what happens when the initial sheaf is unstable; namely, that the limiting behavior should be controlled by the Harder-Narasimhan filtration of the sheaf. In this talk I will describe the solution to this question, which draws on the work of several authors.

Recent progress in automorphism groups

When: Tue, February 3, 2015 - 4:30pm
Where: Math 3206
Speaker: Bingyuan Liu (Washington University) -
Abstract: Let O be a bounded domain of C^n. By a 1935 theorem of Cartan, all biholomorphisms from O onto O form a (real) finite dimensional Lie group, which is denoted by Aut(O). When O is in complex space of one dimension, the study of Aut(O) is classical. However, as one considers domains with higher dimensions,Aut(O) shows both similarity and dissimilarity in terms of algebraic and topological properties comparing with those in one dimension. In this talk, I will give a short introduction and exhibit several recent progresses in the geometry of complex domains with non-compact automorphism groups.

Coniveau and algebraic cycles on very general complete intersections

When: Wed, February 11, 2015 - 4:30pm
Where: (JHU, Shaffer Hall, Room 100)
Speaker: Claire Voisin ((Jussieu/IAS)) -
There are two notions of coniveau (Hodge and geometric) for smooth projective varieties, which should be equivalent according to Grothendieck-Hodge conjecture. On the other hand, the Bloch conjecture is a prediction that Chow groups of algebraic varieties are trivial in dimension smaller than the coniveau. We proved this conjecture for very general complete intersections. The talk will be mainly devoted to explaining the notions and motivating the conjecture mentioned above.

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Scalar-flat Kähler ALE metrics on minimal resolution

When: Tue, March 24, 2015 - 4:30pm
Where: JHU, Gillman Hall 219
Speaker: Michael Lock (University of Texas) -
Abstract: Scalar-flat Kähler ALE surfaces have been studied in a variety of settings since the late 1970s. All previously known examples have group at infinity either cyclic or contained in SU(2). I will describe an existence result for scalar-flat Kähler ALE metrics with group at infinity G, where the underlying space is the minimal resolution of C^2/G, for all finite subgroups G of U(2) which act freely on S^3. I will also discuss a non-existence result for Ricci-flat metrics on certain spaces, which is related to a conjecture of Bando-Kasue-Nakajima. This work is joint with Jeff Viaclovsky.

Connecting toric manifolds by conical Kahler-Einstein metrics

When: Tue, April 7, 2015 - 4:30pm
Where: JHU
Speaker: Ved Datar (University of Notre Dame ) -
Abstract: Conical Kahler-Einstein metrics have played an important role in the recent breakthrough on the existence of smooth Kahler-Einstein metrics on Fano manifolds. In this talk, I will first show that a log-Fano toric pair has conical KE metric if and only if the barycenter of the corresponding moment polytope is zero. This confirms the log version of the Yau-Tian-Donaldson conjecture for toric pairs, and extends a fundamental result of Wang and Zhu. I will then show that any two toric manifolds of the same dimension can be connected by a continuous family of toric manifolds paired with conical Kahler-Einstein metrics in the Gromov-Hausdorff topology. This is joint work with Bin Guo, Jian Song and Xiaowei Wang.

On the convexity of the Mabuchi energy functional along geodesics

When: Tue, April 7, 2015 - 5:30pm
Where: JHU
Speaker: Long Li (McMaster University) -
Abstract: It is conjectured by X.X. Chen that the Mabuchi energy functional is convex along the geodesic connecting two Kaehler metrics, during his study in uniqueness of csck metrics. Now we can give an affirmative answer to this question in the joint work with X.X. Chen and Mihai Paun. The first breakthrough in this subject is the work by Berman and Berndtsson last year, where they proved the weak convexity of the Mabuchi energy functional based on the log-subharmonicity of Bergman kernels. Our work is somewhat a "global version" of Bergman kernels approximation, and also completes the conjecture by proving the continuity of the Mabuchi energy functional along the geodesic. Finally, we also calculated the almost convexity of the Mabuchi energy functional along the \ep-approximation geodesics.

On Bott-Chern Forms of Hermitian Holomorphic Vector Bundles

When: Tue, April 21, 2015 - 4:30pm
Where: JHU, Shaffer Hall, Room 302)
Speaker: Leon Takhtajan (Stony Brook) -
Abstract: : I will discuss a method for explicit computation of Bott-Chern forms of holomorphic Hermitian vector bundles over a complex manifold. After reviewing the basic properties of Chern and Bott-Chern forms, I will describe descent and ascent equations for explicitly computing these differential forms.