Program

Schedule

  wednesday April 20  thursday April 21 friday April 22 
 08:30 - 09:00 registration  
 09:00 - 10:00 Weinkove 1 Weinkove 2 Weinkove 3
 10:00 - 10:30 coffee break coffee break coffee break
 10:30 - 11:30 Otiman Pontecorvo Lock
 11:30 - 12:00 Stanciu Tardini 
 12:00 - 14:30 lunch break lunch break lunch break
 14:30 - 15:30Ornea Apostolov 
 15:30 - 16:00 coffee break coffee break 
 16:00 - 17:00 Moroianu Fino 
20:00 - 22:00  social dinner 

Course

Ben Weinkove, "The complex Monge-Ampère equation on non-Kähler manifolds"

In this mini-course I will describe recent progress on complex Monge-Ampere equations on non-Kähler manifolds and applications to existence of special metrics.  I will also include some details of the proofs and techniques.  The mini-course is intended for a broad audience.  In my first lecture I will discuss some background and motivation, and give an overview of the latest developments in this area.  In the second lecture I will begin a discussion of the a priori estimates required to prove existence of solutions to these equations in the non-Kähler setting, starting with the "zero order" estimate.  In the third lecture, I will talk about the key second order estimates for Monge-Ampère equations, and describe the difficulties arising from the non-Kähler torsion terms. Finally, I'll discuss some open problems in the field.

Talks

Vestislav Apostolov, "Locally Conformally Symplectic Structures on Compact Non-Kaehler Complex Surfaces"

In this talk I will show that every compact complex surface with odd first Betti number admits a locally conformally symplectic 2-form which tames the underlying almost complex structure. I will then discuss a few possible ramifications of this result. This is a joint work with Georges Dloussky.

Anna Fino, "Special hermitian metrics in symplectic geometry"

Symplectic forms taming complex structures on compact manifolds are strictly related to a special type of Hermitian metrics, known in the literature also as "pluriclosed" metrics.
I will present some general results on "pluriclosed" metrics and their link with symplectic geometry on compact solvmanifolds. Moreover, I will show for certain 4-dimensional non-Kaehler 4-manifolds some recent results about the Calabi-Yau problem in the context of symplectic geometry.

Mike Lock, "Special Hermitian metrics characterized by relationships between scalar curvatures"

On a Kahler manifold there is a clear connection between the  complex geometry and underlying Riemannian geometry, which can be used to characterize the Kahler condition. While such a link is not as  clear in the non-Kahler setting, one can seek to understand these  characterizations as specific instances of a more general type.  I will address such questions from the perspective  of relationships between the Chern and Riemannian scalar curvatures. This is joint work with Michael Dabkowski.

Andrei Moroianu, "The holonomy problem for locally conformally Kähler metrics"

A locally conformally Kähler (lcK) manifold is a complex manifold $(M,J)$ together with a $J$-compatible Riemannian metric $g$ which has the property that around every point of $M$ there exists a locally defi.ned Kähler metric belonging to the conformal class of $g$. In this talk I will explain the classification of compact lcK manifolds with reduced holonomy obtained in collaboration with Farid Madani and Mihaela Pilca. In particular, I will describe all compact manifolds admitting two non-homothetic Kähler metrics in the same conformal class.

Liviu Ornea, "On the rank of locally conformally Kaehler manifolds"

I shall review the notion of LCK rank, with focus on the rank of Vaisman and locally conformally Kaehler manifolds with potential. Joint work with Misha Verbitsky.

Alexandra Otiman, "Constructions in locally conformally symplectic geometry"

We present a new construction of LCS manifolds based on the coupling form introduced by Sternberg and Weinstein.

Massimiliano Pontecorvo, "Bi-Hermitian metrics on Kato surfaces"

We discuss contructions of bi-Hermitian metrics on non-Kähler minimal surfaces with positive second Betti number. We will also describe some "topological" obstructions to existence of such metrics on Kato surfaces.

Short talks

Miron Stanciu, "Constructions involving the blow-up of locally conformally symplectic manifolds"

We present a few generalizations of known results in symplectic geometry which provide new methods of constructing locally conformally symplectic (LCS) manifolds, via the process of blowing-up.

Nicoletta Tardini, "Geometrically Bott-Chern formal metrics"

An important result of Deligne, Griffiths, Morgan and Sullivan is that any compact Kähler manifold is formal. In this talk we will discuss a geometric notion of formality related to the Bott-Chern cohomology of a complex manifold. In particular, we will focus on the triple Aeppli-Bott-Chern-Massey products and we will see some explicit examples.
Finally we will show that the existence of geometrically Bott-Chern formal metrics is not an open property under small deformations of the complex structure.
This is a joint work with Adriano Tomassini.