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 coclosed. Typically examples of balanced manifolds are given by
modifications of Kahler
manifolds, twistor spaces over antiselfdual oriented Riemannian 4manifolds
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 BottChern cohomology class of the initial structure and in the Kahler
case reduces to the classical
Calabi flow. The other flow still preserves the BottChern 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 wellposedness and the stability
around KahlerEinstein
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 nonexistence 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: AlmostHermitian metrics of constant Chern scalar curvature.
Abstract: On an almostHermitian manifold, the Chern connection is the unique
connection preserving the almostHermitian structure and having
Jantiinvariant torsion. It has the property that the (0,1)part corresponds
to the CauchyRiemann operator. In this talk, I will compare first the Chern
scalar curvature to the Riemannian scalar curvature induced by the LeviCivita
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 Kontsevichtype recursions for counts of real
curves.
Abstract: Kontsevich's recursion, proved by RuanTian 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 Kontsevichtype recursions for
them in 2007, along with an outline for adapting RuanTian'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 reinterpreting his disk counts as degrees of
relatively oriented pseudocycles from moduli spaces of stable real maps and
lifting cobordisms from DeligneMumford moduli spaces of stable real curves.

10/18: Prof. Mehdi Lejmi (BCC CUNY)
Title: Title: AlmostHermitian 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 almostHermitian manifold, the Chern connection is
the unique connection preserving the almostHermitian structure and having
Jantiinvariant torsion. It has the property that the (0,1)part corresponds
to the CauchyRiemann operator. In this talk, I will compare first the Chern
scalar curvature to the Riemannian scalar curvature induced by the LeviCivita
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 KahlerRicci 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 nonKahler 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: Antiselfdual metrics, twistors, and plane conics.
Abstract:
The twistor space of a Riemannian, oriented 4manifold (M,g) is defined as the
unit sphere bundle of the bundle of selfdual 2forms. This 6manifold, call
it Z, admits a tautological almostcomplex structure, whose integrability is
related to the vanishing of a component of the Riemannian curvature tensor of
(M,g), the selfdual Weyl tensor. Metrics whose selfdual Weyl tensor vanish
are called antiselfdual, 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 4manifold 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 antiselfduality condition, a nonlinear 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
Abstract:
Spring 2019
Fall 2018
Spring 2018
Fall 2017
