Kähler and non-Archimedean geometry in Lyon

Speakers

List of speakers:

Hamid Abban, University of Nottingham
Thibaut Delcroix, Institut Montpelliérain Alexander Grothendieck
Rémi Delloque, Université de Bretagne Occidentale
Ruadhaí Dervan, University of Glasgow
Siarhei Finski, École Polytechnique
Louis Ioos, Université CY Cergy Paris
King Leung Lee, Institut Montpelliérain Alexander Grothendieck
Tran-Trung Nghiem, Institut Camille Jordan UCBL
Annamaria Ortu, Chalmers University of Technology
Léonard Pille-Schneider, University of Regensburg
Lars Martin Sektnan, Chalmers University of Technology

Abstracts

Hamid Abban: A pointless approach to K-stability
K-stability is a notion initially introduced to detect existence of Kähler-Einstein metrics on Fano manifolds. However, the notion proved fruitful beyond this by providing the correct platform to construct compact moduli spaces for Fano varieties over the complex numbers, amongst many other applications. In this talk I will uncover another facet of K-stability by exploring connections to existence of rational points over subfields of the complex numbers. This is based on a joint work with Ivan Cheltsov, Takashi Kishimoto, and Frederic Mangolte.

Thibaut Delcroix: Weight sensitivity in K-stability of Fano varieties
For Fano varieties, weighted K-stability is a variant of K-stability encoding the existence of weighted solitons (e.g. Kähler-Ricci solitons). A remarkable property of toric varieties is that their weighted K-polystability is fully encoded by the vanishing of their weighted Futaki invariant. I will discuss examples and counterexamples of this property for more general threefolds, focusing on spherical Fano threefolds.

Rémi Delloque: Continuity of Hermitian Yang-Mills connections with variations of the metric
We work on a Hermitian compact complex manifold (X,g) of dimension n whose associated 2-form is balanced, which is a slightly weaker condition than the 2-form being closed. In this context, a holomorphic Hermitian vector bundle (E,h) admits a Hermitian Yang-Mills connection if and only if it verifies an algebraic condition of stability. This important result is the Kobayashi-Hitchin correspondence. Stability depends on the class of the balanced metric g on X. During this talk, we try to understand the set of all balanced classes on X for which E is stable and more importantly, the behaviour of the associated Hermitian Yang-Mills connection. We show a continuity result when the class approaches a limit class for which E is destabilised.

Ruadhaí Dervan:  Extremal Kähler metrics on blowups
An old result of Arezzo-Pacard states that if a compact complex manifold without nontrivial holomorphic vector fields admits a constant scalar curvature Kähler metric, then its blowup at any point also admits such metrics in Kähler classes making the exceptional divisor small. When the manifold admits holomorphic vector fields, then there are obstructions related to (algebro-geometric) stability. Székelyhidi has given a complete characterisation of when the blowup admits such metrics in dimension at least three. I will describe recent joint work with Lars Sektnan, wherein we give a complete characterisation in all dimensions that applies also to the setting of extremal Kähler metrics, by using a new, more geometric strategy.  

Siarhei Finski: Kobayashi-Hitchin correspondence for polarized fibrations
We extend the Kobayashi-Hitchin correspondence to general fibrations beyond holomorphic vector bundles. Specifically, for a polarized family of complex projective manifolds, we examine the so-called Wess-Zumino-Witten (WZW) equation, which specializes to the Hermite-Einstein equation, when the fibration is associated with the projectivization of a holomorphic vector bundle. We establish that the existence of approximate solutions to this equation is equivalent to the asymptotic semistability of the direct image sheaves associated with high tensor powers of the polarizing line bundle. Furthermore, we provide a refinement of this correspondence, establishing the sharp lower bound for Yang-Mills-type functional in the framework of the WZW equation.

Louis Ioos: Asymptotics of unitary matrix elements on the spaces of homogeneous polynomials
In this talk, we will show how to estimate the matrix elements of the unitary group acting on the space of homogeneous polynomials as the degree tends to infinity. We will then explain how to obtain this result as a consequence of the off-diagonal asymptotic expansion of the Bergman kernel established by Ma and Marinescu.

King Leung Lee: Chow stability of λ stable toric varieties
In classical Geometric invariant theory (GIT), in order to construct moduli space of projective varieties, we need the varieties to be asymptotically Chow semistable. While it is known that any asymptotically Chow semistable variety is K semistable with vanishing Futaki invariants (for the Chow stability), the converse is not true if the variety is not smooth. In general, asymptotic Chow stability is hard to check. But using a result from Ono, it is possible to compute if a toric variety is asymptotically Chow semistable based on its moment polytope. In this talk, we will provide some sufficient criteria when a toric variety is asymptotically Chow semistable(indeed polystable), and as a consequence, we show that if a K semistable projective toric manifold is K semistable and the Futaki-Ono invariant vanishes, then it is asymptotically Chow polystable. This is a joint work with Professor Naoto Yotsutani.

Tran-Trung Nghiem:  Calabi Yau metrics on complex symmetric spaces
On complex symmetric spaces of rank one, Stenzel constructed explicit examples of Calabi-Yau metrics with smooth (outside the vertex) asymptotic cone. A new feature in higher rank is that the possible candidates for asymptotic cones (which can be classified using algebraic geometry) are generally singular. After an introduction and survey of known results, I will present an existence theorem of Calabi-Yau metrics on complex symmetric spaces of rank two with singular asymptotic cone. This provides new examples of non-compact Calabi-Yau manifolds with irregular asymptotic cone besides the only known example of Conlon-Hein, and covers the rank two symmetric spaces left by Biquard-Delcroix. If time allows, I will also briefly mention a joint work with R. Conlon, providing new Calabi-Yau metrics with smooth toric irregular asymptotic cones, a behavior that cannot arise on complex symmetric spaces.

Annamaria Ortu: Stability and moment maps
A guiding problem in Kähler geometry is to find an equivalence between the existence of a given special metric and an algebro-geometric stability condition. In this context, a natural step towards proving the equivalence is to show that, given a manifold that admits a special metric, stable deformations of the manifold still admit a special metric. I will present a new technique to address this type of problem that relies on reducing to a finite-dimensional problem and on the theory of moment maps.

Léonard Pille-Schneider:  The SYZ conjecture for hypersurfaces
Let X -> D* be a polarized family of Calabi-Yau manifolds, whose complex structure degenerates in the worst possible way. The SYZ conjecture predicts the behavior of the fibers X_t, endowed with their Ricci-flat Kähler metric, as t ->0, and in particular the program of Kontsevich and Soibelman relates it to the non-archimedean analytification of X, viewed as a variety over the non-archimedean field of complex Laurent series. I will explain the ideas of this program and some progress in the case of hypersurfaces.

Lars Martin Sektnan: Constant scalar curvature Kähler metrics and semistable vector bundles
A central question in Kähler geometry is if a Kähler manifold admits a canonical metric, such as a Kähler-Einstein metric or more generally a constant scalar curvature Kähler (cscK) metric, in a given Kähler class. The Yau-Tian-Donaldson conjecture predicts that this is equivalent to an algebraic notion of stability. In this talk, I will discuss a necessary and sufficient condition for the projectivisation of a slope semistable vector bundle to admit cscK metrics in adiabatic classes, when the base admits a cscK metric. In particular, this shows that the existence of cscK metrics is equivalent to K-stability in this setting. Moreover, our construction reduces K-stability to a finite dimensional criterion in terms of intersection numbers associated to the vector bundle. This is joint work with Annamaria Ortu.