Almost Complex Geometry Seminar

Department of Mathematics
The Graduate Center of CUNY

Fridays 11:45am - 1:45pm
Room 3212
Organizers: Luis Fernandez, Scott Wilson Qian Chen

Scope: this seminar is devoted to all topics related to almost complex manifolds, including but not limited to: complex manifolds, symplectic topology and geometry, (almost) Kahler 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 foundational material and present day research in a relaxed format that includes traditional lectures with an actively engaged audience.
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Fall 2019:

8/23: Prof. Luigi Vezzoni (U. di Torino)
Title: Geometric flows of Balanced metrics
Note: Room 3309 for this meeting

Abstract: An Hermitian metric g on a complex manifold (M, J) is called balanced if its fundamental form is co-closed. Typically examples of balanced manifolds are given by modifications of Kahler manifolds, twistor spaces over anti-self-dual oriented Riemannian 4-manifolds and nilmanifolds. In the talk it will be discussed two new geometric flows of balanced structures. The first of them was introduced in [1] and consists in a generalisation of the Calabi flow to the balanced context. The flow preserves the Bott-Chern cohomology class of the initial structure and in the Kahler case reduces to the classical Calabi flow. The other flow still preserves the Bott-Chern class of the initial structure but, in contrast with the first one, it is a potential flow and it does not preserve the Kahler condition. For the both flows it will be discussed the well-posedness and the stability around Kahler-Einstein metrics.

8/30: Organizational meeting
Faculty and students welcome

Abstract: We will make a plan on topics to cover, as well as potential papers to read and work through on days when no speakers are scheduled. Come with questions and/or ideas.

9/6: Prof. Scott Wilson
Title: Harmonic symmetries for Hermitian manifolds

Abstract: I will report on some recent work that generalizes the symmetries of the Hodge numbers of Kahler manifolds to the more general setting of compact complex manifolds with compatible metric. I will review some introductory material and the talk will be self contained. New attendees and students are especially welcome. Here is a preprint: arxiv:1906.02952.

9/13: Prof. Luis Fernandez (BCC CUNY)
Title: Basics on almost complex manifolds

Abstract: Given that some of the regular attendees to the seminar will be away, I will do a general review of basic facts about almost complex, complex, almost Kahler, and Kahler manifolds, including metrics, connections, and questions about integrability. I will look at some particular examples of these objects in some detail (especially the 6 sphere). The talk is intended for a general audience with some knowledge of (real) differential geometry. Students are very welcome.

9/20: Santiago Simanca
Title: Elementary Almost Hermitian Properties of Products of Spheres

Abstract: We consider products of two odd dimensional spheres, or S^2 x S^2, S^2 x S^4, S^2 x S^6, and S^6 x S^6, with suitable almost complex structures. In each case, we describe families of conformal classes of almost Hermitian metrics relative to them, and some tensorial quantities of relevance associated such structures. In the latter three cases, we show that with the metrics we use, these Riemannian manifolds do not carry integrable orthogonal almost complex structures.

9/27: Prof. Zhixu Su and Aleksandar Milivojevic
Title: Smooth or almost complex manifolds with prescribed Betti numbers

Abstract: Prescribing a sequence of Betti numbers (or more specifically the rational homotopy type), is there any almost complex manifold realizing the algebraic data? We will discuss the most basic nontrivial case where the sum of Betti numbers is three. Rational surgery reduces the problem to finding characteristic numbers satisfying certain integrality conditions. We will firstly discuss the smooth case, and then prove the non-existence of any almost complex structure on such a manifold if the dimension is greater than 4. We will also provide a general realization theorem for the almost complex case.

10/4: Prof. Mehdi Lejmi (BCC CUNY)
Title: Almost-Hermitian metrics of constant Chern scalar curvature.

Abstract: On an almost-Hermitian manifold, the Chern connection is the unique connection preserving the almost-Hermitian structure and having J-anti-invariant torsion. It has the property that the (0,1)-part corresponds to the Cauchy-Riemann operator. In this talk, I will compare first the Chern scalar curvature to the Riemannian scalar curvature induced by the Levi-Civita connection. Then, I will discuss an analogue of the Yamabe problem for the Chern scalar curvature. If time permits, I will also discuss other special Hermitian metrics.

10/11: Xujia Chen
Title: Lifting cobordisms and Kontsevich-type recursions for counts of real curves.

Abstract: Kontsevich's recursion, proved by Ruan-Tian in the early 90s, is a recursion formula for the counts of rational holomorphic curves in complex manifolds. For complex fourfolds and sixfolds with a real structure (i.e. a conjugation), signed invariant counts of real rational holomorphic curves were defined by Welschinger in 2003. Solomon interpreted Welschinger's invariants as holomorphic disk counts in 2006 and proposed Kontsevich-type recursions for them in 2007, along with an outline for adapting Ruan-Tian's homotopy style argument to the real setting. For many symplectic fourfolds and sixfolds, these recursions determine all invariants from basic inputs. We establish Solomon's recursions by re-interpreting his disk counts as degrees of relatively oriented pseudocycles from moduli spaces of stable real maps and lifting cobordisms from Deligne-Mumford moduli spaces of stable real curves.

10/18: Prof. Mehdi Lejmi (BCC CUNY)
Title: Title: Almost-Hermitian metrics of constant Chern scalar curvature, part II.

Abstract: This will be a continuation and completion of the talk on 10/4. Previous abstract: On an almost-Hermitian manifold, the Chern connection is the unique connection preserving the almost-Hermitian structure and having J-anti-invariant torsion. It has the property that the (0,1)-part corresponds to the Cauchy-Riemann operator. In this talk, I will compare first the Chern scalar curvature to the Riemannian scalar curvature induced by the Levi-Civita connection. Then, I will discuss an analogue of the Yamabe problem for the Chern scalar curvature. If time permits, I will also discuss other special Hermitian metrics.

10/25: Jeffrey Kroll
Title: An application of the Hard Lefschetz Theorem to the moduli space of flat unitary connections on a Kahler manifold

Abstract: The Hard Lefschetz Theorem for compact Kahler manifolds has a generalization which says that, given any hermitian holomorphic vector bundle equipped with a flat Chern connection, multiplication by the Kahler class of the base induces isomorphisms on complimentary dimensional de Rham cohomology groups with values in the flat bundle. In this talk we consider a (smooth) hermitian vector bundle over a closed Kahler manifold, define the moduli space of flat unitary connections, and use the Hard Lefschetz Theorem to prove that the moduli space has a symplectic structure. The main idea is that to each flat unitary connection there is a unique holomorphic structure on the bundle with respect to which the connection becomes the Chern connection. This result generalizes the fact that the moduli space of principle bundles (with reductive structure group) over a closed oriented surface is symplectic, which was first observed by W. Goldman and led to the discovery of Goldman (Lie) brackets on homotopy classes of oriented loops in surfaces.

11/1: no meeting this week

11/8: Prof. Michael Albanese
Title: Connected Sums of Almost Complex Manifolds

Abstract: When is the connected sum of two almost complex manifolds again an almost complex manifold? In the process of answering this question, we will see that a necessary condition arises on the number of summands needed to ensure an almost complex result. We determine those integers p for which the connected sum of p almost complex manifolds is again an almost complex manifold. In order to rule out certain values of p, we provide examples involving the product of two spheres which then lead us to consider which products of two rational homology spheres admit almost complex structures.

11/15: Prof. Yury Ustinovskiy (NYU)
Title: Hermitian Curvature Flow and uniformization problems in complex geometry.

Abstract: In the last decades, geometric flows have been proved to be a powerful analytic tool in classification and uniformization problems in differential topology and geometry. Application of geometric flows (specifically Kahler-Ricci flow) turned out to be particularly fruitful in the context of Kahler geometry. At the same time there are very few efficient analytic methods available in non-Kahler geometry. In this talk we will define Hermitian Curvature Flow on an arbitrary compact complex manifold. We will study its properties and present applications to the classification problem of complex manifolds admitting compatible metric with "semipositive curvature".

11/22: Marlon de Oliveira Gomes
Title: Anti-self-dual metrics, twistors, and plane conics.

Abstract: The twistor space of a Riemannian, oriented 4-manifold (M,g) is defined as the unit sphere bundle of the bundle of self-dual 2-forms. This 6-manifold, call it Z, admits a tautological almost-complex structure, whose integrability is related to the vanishing of a component of the Riemannian curvature tensor of (M,g), the self-dual Weyl tensor. Metrics whose self-dual Weyl tensor vanish are called anti-self-dual, and the construction of such metrics is the main goal of this talk. To achieve this end, I will briefly describe the Penrose correspondence, which determines which complex manifolds are twistor spaces, and how to obtain the underlying 4-manifold from them. By assuming that the twistor space has enough algebraic functions to define a map to the projective plane, I will discuss how the anti-self-duality condition, a non-linear PDE for a Riemannian metric on M, turns into a linear PDE on the space of conics in the projective plane, and how to obtain continuous families of solutions to the latter using elementary projective geometry.

11/29: no meeting (Thanksgiving break)

12/6: Group discussion
Title: open problems and questions; equivalent notions of integrability.

12/13: problem session and group discussion

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