January 19-20, 2019 · CUNY-City Tech · 285 Jay St, Brooklyn, New York

Organizers: Mehdi Lejmi (CUNY-BCC) and Caner Koca (CUNY-City Tech)

Organizers: Mehdi Lejmi (CUNY-BCC) and Caner Koca (CUNY-City Tech)

Speakers · Abstracts · Registration · Directions · Accommodation · Links

The aim of this workshop is to gather experts in the area of Kähler Geometry and to discuss recent developments in this field. The workshop will take place in the Academic Building of NYC College of Technology (CUNY-City Tech).

This workshop is supported in part by a PSC-CUNY ENHANCED Award (#61768-00 49)

From Einstein‒Maxwell to extremal Calabi Kähler metrics and back

**Abstract**: I will present an equivalence between conformally Einstein‒Maxwell Kähler *4*-manifolds
and extremal Kähler *4*-manifolds in the sense of Calabi. The corresponding pairs of Kähler metrics arise
as transversal Kähler structures of Sasaki metrics on the same CR manifold, and having commuting Sasaki‒Reeb vector
fields. This correspondence extends to higher dimensions using the notion of a weighted extremal Kähler metric
recently studied by Lahdili, and thus illuminates several explicit constructions in Kähler and Sasaki geometry.
It also leads to new existence and non-existence results for extremal Sasaki metrics, suggesting a link between
the notions of relative weighted K-stability of a polarized variety found by Lahdili, and relative K-stability
of the Kähler cone corresponding to a Sasaki polarization, studied by Collins‒Székelyhidi and Boyer‒van Coevering.
This is a joint work with D. Calderbank.

Uniform convexity in \(L^p\) Mabuchi geometry, the space of rays, and geodesic stability

**Abstract**: We show that the \(L^p\) Mabuchi metric spaces are uniformly convex for \(p>1\),
implying that these spaces are uniquely geodesic. Using this result we describe the metric geometry
of \(L^p\) Mabuchi geodesic rays associated to a Kähler manifold. Using the relative Kolodziej type
estimate for complex Monge-Ampere equations, we point out that any \(L^p\) geodesic ray can be approximated
by rays of bounded potentials, with converging radial K-energy. Finally, we use these results to verify
(the uniform version of) Donaldson's geodesic stability conjecture for rays of bounded potentials.
This is joint work with C.H. Lu.

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Compact algebraic compactifications of Hermitian-Yang-Mills moduli space

**Abstract**: A key aspect of gauge theory is finding a suitable compactification for the moduli
space of instantons. For higher dimensional manifolds posessing certain additional geometric structures,
Tian has defined a notion of instanton and made progress towards a compactification analogous to
Uhlenbeck's compactification of the moduli space of anti-self-dual connections on a four-manifold.
In the case when the base manifold is Kähler, and the bundle in question is hermitian, instantons which
are unitary and give rise to a holomorphic structures are Hermitian-Yang-Mills connections. A sequence of
such connections is known to bubble at most along a codimension 2 analytic subvariety, and so one might
hope that there is a gauge theoretic compactification which has the structure of a complex analytic space.
I will attempt to explain why this true in the case when the base is projective. This gives a higher
dimensional analogue of a theorem of Jun Li for algebraic surfaces. This is joint work with Daniel Greb,
Matei Toma, and Richard Wentworth.

Kähler metrics with special geometries on a Koiso-Sakane Manifold

**Abstract**: In this talk I will discuss various nice admissible Kähler metrics
on the three dimensional complex manifold
$$M=P({\mathcal O} \oplus {\mathcal O}(1,-1))\rightarrow {\mathbb C}{\mathbb P}^1 \times {\mathbb C}{\mathbb P}^1.$$
By the famous result of Koiso and Sakane, \(M\) admits a Kähler-Einstein metric and thus \(M\) could appropriately be
called a * Koiso-Sakane manifold*. Actually, due to work by Hwang and Guan, every Kähler class on \(M\) admits
extremal Kähler metrics, as defined by Calabi. As is seen in older work with Apostolov, Gauduchon, and Calderbank,
the Kähler classes for which the Futaki invariant vanishes, and hence the extremal metric have constant scalar curvature
(CSC), are explicitly determined.

On \(M\) (and perhaps some mild generalizations of \(M\)), I will discuss
the existence of *weighted extremal metrics* (Apostolov, Calderbank,
Gauduchon, Legendre, Maschler, and Lahdili).
This part of the talk is based on recent work with Apostolov and Maschler.

Time permitting, I will also talk a little bit about the *\(c\)-projective equivalency* (originally
defined by Otsuki and Tashiro) that occurs between some of the CSC Kähler metrics on \(M\).
This part of the talk is based on recent work with Boyer and Calderbank.

Click **here** to register. There is no registration fee.

Conference Venue (NYCCT Academic Building)

The Tillary Hotel - Brooklyn

From JFK Airport

Take the Airtrain to Jamaica Station. Then take the LIRR to Penn Station. From Penn Station take the A or C line to get off at Jay Street-Metrotech station. The NYCCT Academic Building is a couple of blocks north.

From Newark Airport

Take the Airtrain to Newark Liberty International Airport Train Station. Then take the Northeast Coridor train to Penn Station. From Penn Station take the A or C line to get off at Jay Street-Metrotech station. The NYCCT Academic Building is a couple of blocks north.

From LaGuardia Airport

Take the Q33 Bus to Roosevelt Ave - Jackson Heights Subway Station. Then take the F train, and get off at Jay Street-Metrotech station. The NYCCT Academic Building is a couple of blocks north.

We have blocked a number of rooms for the participants at the **Tillary Hotel-Brooklyn **at a special rate ($129/night) for conference participants.
The hotel is right next to the conference venue (NYCCT Academic Building).

You can book your room at this special rate by clicking here. Alternatively, you may call the Tillary Hotel to reserve your room. Please mention "CUNY Math Conference" for reservation.

**Hotel Information:** The Tillary Hotel, 85 Flatbush Ave Ext, Brooklyn, NY 11201. +1 718-329-9537