MARGE 4
New directions in complex geometry

March 9

15:30 Guilherme Cerqueira Gonçalves

Title: Singular Gauduchon Conjecture

Abstract: In 1984 Gauduchon conjectured that one can find Gauduchon metrics with prescribed Ricci curvature on all compact complex manifolds. This conjecture was settled by Székelyhidi-Tosatti-Weinkove in 2019. In this talk I will present a singular version of this conjecture for degenerating families and discuss a few first results towards its solution. I will describe the (n-1) Monge-Ampère equation, fully non-linear elliptic PDE at the heart of this problem. As time allows I will prove that the potential for a degenerate (n-1) Monge-Ampère equation on a compact Hermitian smoothable variety with canonical isolated singularities is uniformly bounded and that on the holomorphic deformation of a Kähler Calabi-Yau manifold we can construct a non-Kähler Calabi-Yau-Gauduchon metrics, inside fixed Gauduchon classes, with uniform bounds independent of the complex structure.

March 10

09:00 Café

09:30 Yuetong Fang

Title: Continuity of solutions to the complex Hessian equations on compact Hermitian manifolds

Abstract: On a Hermitian manifold, we study complex Hessian equations, which interpolate between the Laplace equation and the complex Monge–Ampère equation. In 1998, Kołodziej extended Yau’s C^0 estimate for the complex Monge–Ampère equation to the case where the right-hand side density belongs to certain Orlicz spaces, thereby initiating further study of these equations under his optimal integrability condition.

In this talk, I will present some recent advances on the existence of continuous solutions to the complex Hessian equations. In particular, I will explain how to obtain uniform C^0 estimates when the right-hand side lies in some Orlicz spaces, satisfying Kołodziej's integrability condition. As a consequence, under the same assumptions on the density, any bounded solution is in fact continuous.

10:30 Andrés Moreno

Title: Flows of SU(n)-structures

Abstract: As part of a general approach to the study of flows of H-structures, in this talk we discuss the case when the structure group is H=SU(n). In particular, we focus on the negative gradient flow of a natural Dirichlet-type energy functional and on a coupled Ricci harmonic flow of SU(n)-structures. For those flows, we characterize the critical points, we prove short-time existence and uniqueness as well as Shi-type estimates and long-time existence of solutions of the Ricci harmonic flow of SU(n)-structures. This is a joint work with U. Fowdar (Univ. Warsaw), E. Loubeau (Univ. Brest) and H. Sá Earp (Unicamp).

13:30 Quentin Peres

Title: A twist construction of SU(n)-structures

Abstract: An SU(n)-structure on a manifold may be thought as a Calabi-Yau structure with torsion. After introducing the notion of SU(n)-structures and the different kind of geometries one can get, I will present a construction of SU(n)-structures on quotient of Kähler manifolds by a hamiltonian action of a compact torus, generalizing a construction by Larfors, Lüst and Tsimpis in their paper of 2010. I will also present an application of this result in the case of Calabi-Yau cones over Fano Kähler-Einstein manifolds recovering the well-known example of CP^3. If time allows, I will also discuss recent advances in the understanding of a special type of SU(n)-structures, called LT structures, from a more geometrical point of view.

14:30 Yifan Chen

Title: When singular Kahler-Einstein metrics are Kahler currents

Abstract: We show that a general class of singular Kahler metrics with Ricci curvature bounded below define Kahler currents. In particular the result applies to singular Kahler-Einstein metrics on klt pairs. If time permits, we would also present the RCD property for the completion of the smooth part of Kahler Einstein metric under further conditions. This is a joint work with Shih-Kai Chiu, Max Hallgren, Gábor Székelyhidi, Tat Dat Tô, and Freid Tong.

16:00 Antonio Trusiani

Title: Convexity of the Mabuchi functional in big cohomology classes

Abstract: The Mabuchi functional for big cohomology classes will be defined. For Kähler classes this functional is the Euler-Lagrange functional of the constant scalar curvature Kähler (cscK) equation and one of its key properties is the convexity along weak geodesics proved by Berman–Berndtsson. For big cohomology classes, I will introduce an invariant asso- ciated to the existence of (transcendental) Fujita approximations with good properties, proving that the vanishing of such invariant gives the convexity of the Mabuchi functional along weak geodesics.

I will provide examples and, time permitting, some pluripotential-theoretical appli- cations will also be presented

March 11

09:00 Informal discussions