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

Group seminar 2021

November 26th

Henrique Sá Earp (Universidade Estadual de Campinas)

Title: Some questions we can solve together under MathAmSud

Abstract: In this talk I will review the background and elaborate on some questions raised in our MathAmSud scientific project, aiming at establishing active collaboration. The first set of questions is intended for the “differential geometry” faction, whereas the second set might interest the “algebraic geometry” gang. Sections 2.1, 2.2 of our project address harmonicity of geometric H-structures, on a fixed Riemannian manifold. In this context, computing the precise harmonicity condition for a given symmetry group H is a laborious but clear task, so I believe members of this collaboration might be interested in finding the precise meaning of harmonicity for their preferred type of structure, and relate it to other conditions affecting torsion in each context (eg. generalised complex structures, Vaisman structures…?) Section 2.4 addresses the construction of G2-instantons over twisted connected sum 7-manifolds, obtained by gluing together two semi-Fano 3-fold building blocks. In previous we have set up a technique to construct suitable instanton bundles over blocks by means of a Hartshorne-Serre correspondence, in which bundles are essentially parametrised by curves. So now can we weaken some of the topological restrictions on the bundles, in order to construct *singular* examples, i.e., reflexive sheaves satisfying an adequate version of the same numerical constraints? Time allowing, we can also talk about cCY 7-manifolds

October 29th

Adrián Andrada (Universidad Nacional de Córdoba-CONICET)

Title: "Locally conformally Kähler and Vaisman structures on compact solvmanifolds"

Abstract: A Hermitian manifold (M,J,g) is called locally conformally Kähler (LCK for short) if each point in M has a neighbourhood where the metric g is conformal to a Kähler metric. This is equivalent to the existence of a closed 1-form \theta satisfying d\omega=\theta\wedge omega, where \omega denotes the fundamental 2-form. When the form \theta is parallel with respect to the Levi-Civita connection, the manifold is called Vaisman. In this talk I will review properties of left invariant LCK and Vaisman structures on solvmanifolds, that is, compact manifolds obtained as the quotient of a simply connected solvable Lie group by a discrete subgroup. We will also analyze the holonomy of the Bismut connection on LCK and Vaisman manifolds. These results were obtained in collaboration with Marcos Origlia (Córdoba) and Raquel Villacampa (Zaragoza).

September 17th:

Charles Almeida (Universidade Federal de Minas Gerais)

Title: "Lefschetz Properties and osculating defectiveness of algebraic varieties."

Abstract: The study and classification of varieties whose osculating space fails to have the expected dimension is a classical and long-standing problem that goes back to the Italian school. The purpose of this talk is to establish a close relationship between this problem and the study of the presence of so-called Lefschetz Properties in artinian algebras.

July 16th:

Francisco Iván Rubilar Arriagada (Universidad Católica del Norte)

Title: "Deformations of noncompact complex manifolds and their moduli of vector bundles"

Abstract: I will present recent developments of deformations of complex structures of noncompact complex manifolds focused on surfaces and Calabi--Yau 3-folds, and their moduli of vector bundles. If time permits, I will present and discuss some open problems concerning deformations and their effect on the moduli of vector bundles.

June 11th:

Daniele Faenzi (Université de Bourgogne)

Title: "Logarithmic sheaves of divisors and complete intersections: freeness and stability"

Abstract:I will give an overview of logarithmic sheaves attached to hypersurfaces and to our recent exploration of the notion of logarithmic sheaves of complete intersections. I would like to discuss briefly divisors and complete intersections having free vs stable logarithmic sheaves. If time allows I will describe the case of determinants and/or pencils of quadrics --- joint work with M. Jardim, S. Marchesi, J. Valles.

May 14th:

Andrés Moreno (UNICAMP)

Title: "Harmonic Sp(2)-invariant G2-structures on the 7-sphere"

Abstract: According to the classification of compact homogeneous spaces with invariant G2-structure, due by Reidegel (2010) and improved by Van Le-Munir (2012), we give an explicit description of the space of homogeneous G2-structures on the 7-sphere, in terms of their isometric classes. Furthermore, we find critical points of a natural Dirichlet energy of G2-structures, characterised by the free divergence torsion condition (div T=0). Also, we prove that some of these critical points are unstable, as well as, they can be realised as limit points of invariant solutions of the corresponding gradient flow. This is joint work with E. Loubeau, H. Sá Earp and J. Saavedra. arXiv:2103.11552v1

April 16th:

Gonçalo Oliveira (UFF-Niteroi)

Title: "Special Lagrangians and Lagrangian mean curvature flow"

Abstract: (joint work with Jason Lotay) A standing conjecture of Richard Thomas, motivated by mirror symmetry, gives a stability condition supposed to control the existence of a special Lagrangian submanifold in a given Hamiltonian isotopy class of Lagrangians. Later, Thomas and Yau conjectured a similar stability condition controls the long-time existence of the Lagrangian mean-curvature flow. In this talk, I will explain how Jason Lotay and myself have recently proved versions of these conjectures on circle symmetric hyperKahler 4-manifolds.