Monge-Ampere equation and Calabi-Yau manifolds:
talks and abstracts

May 23-27, 2015
Laboratory of Algebraic Geometry, Moscow

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Zbigniew Blocki (Krakow)
The Calabi-Yau Theorem

We will sketch the proof of the Calabi-Yau theorem which asserts that on a compact Kahler manifold M the map sending a Kahler metric from a given Kahler class to its Ricci curvature, which is an element of the first Chern class of M, is bijective. It is equivalent to proving that every positive volume form on M with a fixed total volume is a Kahler volume of a unique Kahler metric (in a given Kahler class), and this boils down to solving the complex Monge-Ampere equation. Building up on the work of Calabi, Nirenberg and Aubin, Shing-Tung Yau finished the proof of this result in 1976, for which he was awarded the Fields Medal in 1982. One of the amazing consequences is existence of Ricci-flat metrics on manifolds with vanishing first Chern class. Except for toruses, these metrics can never be written down explicitly!

Literature:

Zbigniew Blocki (Krakow)
The space of Kahler metrics

Mabuchi, Semmes and Donaldson have independently introduced a natural Riemannian structure on (infitely dimensional) space of Kahler metrics on a compact manifold fo dimension n. It turns out that geodesics in this space are solutions of the homogeneous complex Monge-Ampere equation in the space of dimension n+1. X.X.Chen proved existence of weak almost C^{1,1} geodesics and Lempert and Vivas showed that they need not be smooth in general. Nevertheless, these geodesics play crucial role in the proof of uniqueness of constant scalar curvature metrics modulo holomorphic automorphism in a given Kahler class.

Slawomir Dinew (Krakow)
Local regularity of the complex Monge-Ampere equation

We shall review recent results on the local regularity of the complex Monge-Ampere equation. In particular we shall be interested in the problem of optimal conditions implying that local solutions of the equation with "good" data are smooth. Several illustrating examples will be shown. We shall also compare the existent results with the analogous real Monge-Ampere theory due to Caffarelli.

Jakob Hultgren (Chalmers)
Coupled Kahler-Einstein Metrics

A central theme in complex geometry has been to study various types of canonical metrics, for example Kahler-Einstein metrics and cscK metrics. In this talk we will introduce the notion of coupled Kahler-Einstein (cKE) metrics which are k-tuples of Kahler metrics that satisfy certain coupled Kahler-Einstein equations. We will discuss existence and uniqueness properties and elaborate on related algebraic stability conditions. (Joint work with David Witt Nystrom)

Mattias Jonsson (Ann Arbor)
On degenerations of volume forms and Calabi-Yau varieties

Various considerations in mirror symmetry and moduli problems leads one to study degenerating one-parameter families of complex manifolds. For example, a conjecture by Kontsevich-Soibelman describes the Gromov-Hausdorff limits of certain degenerations of Calabi-Yau manifolds.

I will give an introduction to these circle of ideas, and then present joint work with Sebastien Boucksom, where we study the asymptotic behavior of volume forms (i.e. smooth measures) on a degenerating family of compact complex manifolds. Under rather general conditions, we prove that the volume forms converge in a natural sense to a Lebesgue-type measure on a certain simplicial complex.

Alexander Kolesnikov (HSE)
Hessian metrics related to mass transportation problem. Applications to geometry and analysis

The Monge-Kantorovich šoptimal mass transporation problem has numerous applications in analysis, probability, and PDE's. We will talk about its interplay with the differential geometry, šfocusing the metric properties of the related Hessian space with convex potential solving the corresponding real Monge-Ampere equation. We discuss recent extensions of some classical results on the Kaehler-Einstein equation and explain thešmotivation coming from convex geometry and probability.š

Slawomir Kolodziej (Krakow)
Pluripotential methods of solving the complex Monge-Ampere equation

The aim of the mini-course is to show how the pluripotential methods can be applied to prove existence and stability of weak solutions of the Monge-Ampere equation on compact Kahler or just Hermitian manifolds. First the basic notions and results of pluripotential theory (after Bedford and Taylor) are discussed. Using them we will sketch the proof of a priori estimates and the existence of solutions of the complex Monge-Ampere equation on Kahler manifolds when the right hand side is in Lp(X, n), p > 1. Then we adapt the methods to the Hermitian case.

For background material see the first chapter of: S. Kolodziej, The complex Monge-Ampere equation and pluripotential theory, Memoirs Amer. Math. Soc. 178 (2005), pp. 64.

Magnus Önnheim (Göteborg)
Monge-Ampere equations on compact Hessian manifold

In the study of complex Monge-Ampere equations, variational formulations have managed to produce several existence and uniqueness results. In the setting of Euclidean space, a variational formulation of the real inhomogeneous Monge-Ampere equation is given by the Kantorovich functional from optimal transport (analogous to the Ding functional in complex geometry).

In this talk, inspired by ideas of Shima and Gross, we show that the Legendre transform can be given a natural geometric meaning in the setting of a Hessian manifold. We proceed to show that using the Legendre transform to define a Kantorovich type functional we obtain a variational formulation for inhomogeneous Monge-Ampere equations on M. One feature of this approach is that it provides a way to handle very singular equations. If time permits we briefly discuss some interesting phenomena relating to this. (Joint work with Jakob Hultgren)

Vladimir Roubtsov (Angers)
Monge-Ampere equations, geometric structures and meteorological hydrodynamics

We describe how the geometric approach to Monge-Ampere operators and equations proposed by V.Lychagin might be used in (quasi-)geostrophic hydrodynamical models. Some naturally arised geometric structures (Kahler, Hyperkahler/symplectic, Generalized Complex...) in this models are discussed.

Jian Song (Rutgers)
The Kahler-Ricci flow through singularities

We study the formation of finite time singularities for the Kahler-Ricci flow in relation to birational surgeries such as divisorial contractions and flips. We show that the Kahler-Ricci flow can be uniquely continued through such birational surgeries. In particular, we construct a global family of flips by the Kahler-Ricci flow in Gromov-Hausdorff topology.

Yuguang Zhang (Beijing)
Balanced embedding of degenerating Abelian varieties

In this talk, I show that, after a base change, a certain degeneration of Abelian varieties can be simultaneously balanced embedded into a projective space by the theta functions. Then I study the relationship between the balanced central fiber and the Gromov-Hausdorff limit of flat Kahler metrics on the nearby fibers.

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Laboratory of Algebraic Geometry and its Applications