MATH AMSUD project: GS&MS 21-MATH-06 – 2021 & 2022

MAS @ IMECC 2022

During the months of March and April, colleagues from Argentina, France, Chile and Brazil will visit the Geometry Group at IMECC, in the framework of our Math-AmSud collaboration.

Here are some details of the visits:

21/03 -- 01/04: Adrián Andrada and Alejandro Tolcachier, Universidad Nacional de Córdoba, Argentina.

28/03 -- 10/04: Gregoire Menet, Université de Lille, France.

28/03 -- 15/04: Bruno Suzuki, Universidad Católica del Norte, Chile.

04/04 -- 08/04: Andrew Clarke, Universidade Federal de Rio de Janeiro, Brazil.

Planned activity:

April 1st, Friday

There will take place 4 talks, at the Auditorium of the Institute:

9:30h Coffee reception.

10h Adrián Andrada

• Title: On non-Kähler complex manifolds with trivial canonical bundle

• Abstract: A complex manifold of complex dimension n has holomorphically trivial canonical bundle if there exists a nowhere vanishing holomorphic (n,0)-form. Recently, much interest has been devoted to studying compact complex manifolds with this property. For instance, any compact nilmanifold with an invariant complex structure has trivial canonical bundle, and it is not Kähler unless it is a torus. In this talk we will provide new examples of this kind of complex manifolds: first, among solvmanifolds and second, among complex manifolds obtained as the product of two normal almost contact manifolds. Joint work with M. Origlia and A. Tolcachier.

11h Alejandro Tolcachier

• Title: Holonomy groups of flat solvmanifolds and relations with G_2-geometry

• Abstract: A solvmanifold is defined as a compact quotient of a simply-connected solvable Lie group by a discrete subgroup. In this talk we will consider flat solvmanifolds, i.e. solvmanifolds endowed with a flat Riemannian metric induced by a flat left invariant metric on the associated Lie group. Milnor gave a characterization of Lie groups which admit a flat left invariant metric and he showed that they are all solvable of a very restricted form. On the other hand, the fundamental group of a flat (solv)manifold is a Bieberbach group. In particular, it admits a free abelian normal subgroup of finite index, and the holonomy group of the flat manifold is finite. Using these tools we will prove some properties of the holonomy group. Namely, it is abelian and conversely every finite abelian group is the holonomy group of a flat solvmanifold. We will briefly comment on the classification of flat solvmanifolds in dimensions up to 6, which is related to the problem of determining conjugacy classes of subgroups of GL(n,Z). Finally, we will talk about the existence of closed and coclosed G_2 structures on 7-dimensional flat solvmanifolds.

12h Lunch.

14h Gregoire Menet

• Title: Construction of hyperkähler orbifolds

• Abstract: In order to classify the complex varieties, hyperkähler manifolds have appeared as important objects because of their role of elementary bricks in the Bogomolov decomposition theorem. More recently this theory has been generalized to the singular setting. In this context, the hyperkähler orbifolds can be seen as one of the simplest generalization of the hyperkähler manifolds with only quotient singularities. In this talk, we will see how to construct examples of such orbifolds.

15:00 Coffee break.

15:30h Bruno Suzuki

• Title: Deformations of Calabi-Yau threefolds

• Abstract: We adapt the Kodaira theory of deformations of compact complex manifolds in order to compute deformations of noncompact Calabi--Yau threefolds, obtaining infinite dimensional deformation families.We then study the geometry of these deformations.