Summary of the talks.
Speaker: Frank Loray.
Title: Deligne-Mumford semi-compactification of Painlevé VI equation.
Date: March 6, 2023 - 11 hs. Sala 407 - Bloco H - Campus do Gragoatá UFF.
Abstract:
In a work in progress with Gabriel Calsamiglia (UFF, Niteroi, Brasil) and Titouan Serandour (Univ Rennes/ENS Lyon) we investigate a partial compactification of the phase space of Painlev´e VI equations by using Deligne-Mumford compactification of the moduli space of the 5-punctured sphere. Our motivation is to understand in a geometric way the asymptotics of Painlevé VI transcendents.
Speaker: Jorge Vitório Pereira .
Title: Submanifolds with ample normal bundle.
Date: March 6, 2023 - 14 hs. Sala 407 - Bloco H - Campus do Gragoatá UFF.
Speaker: João Paulo Figueredo.
Title: Birational geometry of holomorphic foliations.
Date: March 6, 2023 - 15 hs. Sala 407 - Bloco H - Campus do Gragoatá UFF.
Abstract:
I will start this talk by going over some basic facts about the birational geometry of holomorphic foliations, for instance the birational classification of foliations on surfaces and the Minimal Model Program for foliations on varieties of dimension at most three. Finally, I will go over some aspects of Fano foliations and the classification of the ones having large index.
Speaker: Paulo Sad.
Title: Tipo de Ueda 1 e folheacoes.
Date: March 7, 2023 - 11 hs. Sala 407 - Bloco H - Campus do Gragoatá UFF.
Abstract:
Sao exibidos exemplos de folheacoes (adaptadas ao exemplo de Neeman) possuindo uma folha compacta.
Speaker: Sebastián Velazquez.
Title: Stability of pullbacks of foliations on weighted projective spaces.
Date: March 7, 2023 - 14 hs. Sala 407 - Bloco H - Campus do Gragoatá UFF.
Abstract:
In this talk we will show a stability-type theorem for foliations on projective spaces which arise as pullbacks of foliations on weighted projective spaces. As a consequence, we will be able to construct many irreducible components of the corresponding moduli space of foliations, most of them being previously unknown. This result also provides an alternative and unified proof for the stability of other families of foliations.
Speaker: Frédéric Touzet.
Title: Structure of Kähler foliations with negative transverse Ricci curvature (joint work with Benoît Claudon).
Date: March 7, 2023 - 15 hs. Sala 407 - Bloco H - Campus do Gragoatá UFF.
Abstract:
We investigate the structure of transversely Kähler foliations with quasi-negative tranverse Ricci curvature. In particular, we prove a de Rham type theorem decomposition on the leaf space where we characterize each factor. These statements shall be seen as foliated analogues of results obtained by Nadel and Frankel concerning the geometry of the universal covering $\wtX$ of $X$ a complex projective manifold with ample canonical bundle.
Speaker: Carolina Araujo.
Title: The Calabi Problem for Fano Threefolds.
Date: March 8, 2023 - 14 hs.
Abstract:
A formidable problem in the confluence of differential and algebraic geometry is to determine which complex manifolds admit a Kähler-Einstein metric. A necessary condition for the existence of such a metric is that the canonical class of the manifold has a definite sign. In his 1954 ICM lecture, Calabi introduced the problem and conjectured that if the canonical class of a compact Kähler manifold X has a definite sign, then X should admit a Kähler-Einstein metric. The problem became known as the “Calabi problem”.
The Calabi problem was solved for manifolds with zero and positive canonical class by Yau and Aubin/Yau in the 1970s. They confirmed Calabi’s prediction, showing that these manifolds always admit a Kähler-Einstein metric. (This achievement won Yau the Fields Medal in 1982.) On the other hand, for projective manifolds with positive curvature, called “Fano manifolds”, the Calabi problem can have a negative solution, as first observed by Matsushima in 1957. In the last 30 years, the Calabi problem for Fano manifolds has attracted attention of many algebraic and differential geometers, resulting in the famous Yau-Tian-Donaldson conjecture, which states that a Fano manifold admits a K¨ahler-Einstein metric if and only if it satisfies a sophisticated algebraic-geometric condition, called “K-polystability”. In 2012, this conjecture was solved by Chen, Donaldson and Sun. Since then, birational geometry tools have been usedwith great success to attack the problem.
In this talk, I will present an overview of the Calabi problem, describing some of the important recent developments in connection with birational geometry.
Speaker: Gabriel Calsamiglia.
Title: Isoperiodic foliations in moduli spaces of mermorphic forms with simple poles.
Date: March 8, 2023 - 15 hs. Sala 407 - Bloco H - Campus do Gragoatá UFF.
Abstract:
The equivalence relation ”having the same periods” defines a regular holomorphic foliation in the moduli space of meromorphic differentials with at worst simple poles on complex curves of fixed genus and number of poles. We will present some results that allow to describe the closure of each leaf, in terms of the topological properties of the set of periods in the complex plane.
As a consequence of the techniques, we will deduce that, in the case of two simple poles, there are an infinite number of closed leaves that are algebraic (considered as subsets of the Deligne-Mumford compactification of the fibered ambient space) and also an infinite number of leaves that are closed, but not algebraic .