## 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 - 15:30* | Ornea | 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. |