CUNY Kähler Geometry Workshop

Vestislav Apostolov

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.

Tamas Darvas

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 and novel Laplacian estimates, we point out that any \(L^p\) geodesic ray can be approximated by rays having bounded Laplacian, with converging radial K-energy. We use these results to verify (the uniform version of) Donaldson's geodesic stability conjecture for rays with bounded Laplacian. This is joint work with C.H. Lu.

Weiyong He

Abstract: Recently Xiuxiong Chen and Jingrui Cheng have made a breakthrough on the existence of constant scalar curvature metrics on compact Kähler manifolds, in view of Calabi-Donaldson program and Yau-Tian-Donaldson conjecture.

The essential new input is a highly nontrivial a priori estimates for scalar curvature type equation, which is a fully nonlinear fourth order elliptic PDE. We will discuss the exciting developments regarding the existence of constant scalar curvature metrics and their extensions.

Hongnian Huang

Abstract: In Kähler geometry, an interesting question is that how we can smooth a weak Kähler metric. A lot of techniques have been developed to smooth a weak Kähler metric, e.g., the recent breakthrough of Chen-Cheng's work. In this talk, we will discuss how to use the Calabi flow to smooth weak Kähler metrics on an abelian surface.

Claude LeBrun

Abstract: In the author's previous joint work with Hans-Joachim Hein, a mass formula for asymptotically locally Euclidean (ALE) Kähler manifolds was proved, assuming only relatively weak fall-off conditions on the metric. However, the case of real dimension \(4\) presented technical difficulties that led us to require fall-off conditions in this special dimension that are stronger than the Chruściel fall-off conditions that sufficed in higher dimensions. In this talk, I will explain how a new proof of the \(4\)-dimensional case, using ideas from symplectic geometry, shows that Chruściel fall-off suffices to imply all our main results in any dimension. In particular, I will explain why our Penrose-type inequality for the mass of an asymptotically Euclidean Kähler manifold always still holds, given only this very weak metric fall-off hypothesis.

Lars Sektnan

Abstract: Metrics of Poincaré type are Kähler metrics defined on the complement \(X\backslash D\) of a smooth divisor D in a compact Kähler manifold \(X\) which near \(D\) are modeled on the product of a smooth metric on D with the standard cusp metric on a punctured disk in \(\mathbb{C}^n\). In this talk I will discuss an Arezzo-Pacard type theorem for such metrics. A key feature is an obstruction which has no analogue in the compact case, coming from additional cokernel elements for the linearisation of the scalar curvature operator. This condition is conjecturally related to ensuring the metrics remain of Poincaré type.

Ben Sibley

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.

Christina Tonnesen-Friedman

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.