Almost Complex Geometry Seminar

Department of Mathematics
The Graduate Center of CUNY

Fridays 12noon - 1:30pm
via Zoom, link below, open to public
Organizers: Luis Fernandez, Mehdi Lejmi Scott Wilson

Scope: this seminar is devoted to all topics related to almost complex manifolds, including complex manifolds, symplectic topology and geometry, (almost) Kahler geometry, and spin_c geometry, as well the tools of algebraic topology and geometric analysis that have proven useful in studying such structures. The goal is expose students and faculty to present day research and foundational material in a relaxed format that includes traditional lectures with an actively engaged audience.
Prior Semesters: Fall 2020 Fall 2019 Spring 2019 Fall 2018 Spring 2018 Fall 2017

Spring 2021:

Join Zoom Meeting
Meeting ID: 825 6324 2686
Passcode: 999054

2/5: Prof. Teng Fei (Rutgers-Newark)
Title: A geometric flow for Type IIA superstrings

Abstract: The equations of flux compactifications of Type IIA superstrings were written down by Tomasiello and Tseng-Yau. To study these equations, we introduce a natural geometric flow on symplectic Calabi-Yau 6-manifolds. We prove the well-posedness of this flow and establish the basic estimates. As an application, we make use of our flow to find optimal almost complex structures on certain homogeneous symplectic half-flat manifolds. This is based on joint work with Phong, Picard and Zhang.

2/12: no meeting (University closed)

2/19: Prof. Carlo Scarpa (SISSA)
Title: The Hitchin-cscK equations in symplectic coordinates

Abstract: The Hitchin-cscK system (HcscK, for short) is a pair of PDEs for a Kahler metric and an infinitesimal deformation of the complex structure, that couple the constant scalar curvature equation with a weak holomorphicity condition on the infinitesimal deformation. The system is derived from a hyperkahler extension of the Fujiki-Donaldson interpretation of the scalar curvature as a moment map. In this talk I will review some aspects of this construction, to highlight the similarity with Hitchin's description of the Higgs bundles equations, and I will present some recent results, contained in arxiv:2006.06250, about solutions of the Hitchin-cscK system on toric and abelian varieties. This is joint work with Jacopo Stoppa.

2/26: Prof. Francesco Pediconi (U. Florence)
Title: On cohomogeneity-one Hermitian non-Kahler manifolds

Abstract: In this talk, we will consider Hermitian manifolds acted by a (real) compact Lie group by holomorphic isometries with principal orbit of codimension one. In particular, we will focus on a special class of these manifolds constructed by following Berard-Berger. On such spaces, we characterize the special Hermitian non-Kahler metrics, such as balanced, pluriclosed, locally conformally Kahler, and we provide new examples of inhomogeneous non-Kahler second-Chern-Einstein metrics. This is a joint work with Daniele Angella.

3/5: Prof. Jonas Stelzig (LMU Munchen)
Title: Bott-Chern cohomology, formality and maximally non-integrable structures.

Abstract: In the first part of the talk, I will define promising looking candidates for Bott-Chern and Aeppli cohomology for almost complex manifolds which give rise to a ddbar-type condition that implies formality. In the second part, I will report on work in progress with Giovanni Placini and Rui Coelho, revolving around maximally non-integrable almost complex structures, mostly limiting myself to real dimension 4, where we show an h-principle, giving rise to many examples. A study of the Dolbeault and Bott Chern cohomologies of such structures will lead to a more critical view on the notions introduced in the first half of the talk.

3/12: Prof. Alexandra Otiman (Roma Tre University)
Title: Toric Kato manifolds

Abstract: Kato manifolds are compact complex manifolds containing a global spherical shell. Their modern study has been widely carried out in complex dimension 2 and originates in the seminal work of Inoue, Kato, Nakamura and Hirzebruch. In this talk I plan to describe a special class of Kato manifolds in arbitrary complex dimension, whose construction arises from toric geometry. Using the toric language, I will present several of their analytic and geometric properties, including existence of special complex submanifolds and partial results on their Dolbeault cohomology. Moreover, since they are compact complex manifolds of non-Kahler type, I will investigate what special Hermitian metrics they support. This is joint work with Nicolina Istrati (Marburg), Massimiliano Pontecorvo (Rome) and Matteo Ruggiero (Paris).

3/19: no meeting (postponed to next week)

3/26: Xi Sisi Shen (Northwestern)
Title: Metrics of constant Chern scalar curvature and a Chern-Calabi flow

Abstract: We discuss the existence problem of constant Chern scalar curvature metrics on a compact complex manifold. We prove a priori estimates for these metrics conditional on an upper bound on the entropy, extending a recent result by Chen-Cheng in the Kahler setting. In addition, we show how these estimates can be used to prove a convergence result for a Hermitian analogue of the Calabi flow on compact complex manifolds with vanishing first Bott-Chern class.

4/2: no meeting (spring break)

4/9: Prof. Nicoletta Tardini (University di Parma)
Title: SKT and Kahler-like metrics on complex manifolds.

Abstract: Several special non-Kahler Hermitian metrics can be introduced on complex manifolds. Among them, SKT metrics deserve particular attention. They can be defined on a complex manifold by saying that the torsion of the Bismut connection associated to the metric is closed. These metrics always exist on compact complex surfaces but the situation in higher dimension is very different. We will discuss several properties concerning these metrics also in relation with the Bismut connection having Kahler-like curvature. Since this last property on nilmanifolds will force the complex structure to be abelian, we will also discuss the relation between SKT metrics and abelian complex structures on unimodular Lie algebras. These are joint works with Anna Fino and Luigi Vezzoni.

4/16: Mattia Pujia (U. Torino)
Title: Geometric flows and special Hermitian metrics on Lie groups

Abstract: The talk focuses on geometric flows of Hermitian metrics on Lie groups. In particular, we will present long-time existence and convergence results for the so-called Hermitian curvature flow, which was introduced by Streets and Tian. We will also discuss existence and uniqueness of soliton solutions to this flow, pointing out some algebraic obstructions on the existence of such metrics. This is a joint work with R. Lafuente and L. Vezzoni.

4/23: informal meeting scheduled (all are welcome)

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