Complex Geometry in Bordeaux - Sciencesconf.org

26-29 mai 2026 Bordeaux (France)

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Au sujet de la rencontre

Nous organisons un workshop en géométrie complexe, avec trois mini-cours par Ruadhaí Dervan, Mirko Mauri et Junsheng Zhang, ainsi que deux exposés de recherche.

La rencontre se tiendra du 26 au 29 Mai 2026 à l'Institut de Mathématiques de Bordeaux.

Les participant·es devront en principe financer leur déplacement et leur hébergement, mais nous aurons un peu d'argent pour financer quelques jeunes collègues, qui pourront indiquer dans le formulaire s'ils ou elles souhaitent candidater.

La date limite pour les pré-inscriptions est fixée au 20 février 2026. Les participants acceptés recevront un email dans les jours qui suivront.

Programme

MINI-COURSES

Ruadhai Dervan

Arcs and the constant scalar curvature Kähler metrics

The Yau-Tian-Donaldson conjecture states that there is an algebro-geometric characterisation of the existence of a constant scalar curvature Kähler metric on a projective variety. A solution to this conjecture has recently been given by Boucksom-Jonsson and Darvas-Zhang, via algebro-geometric stability conditions for which we currently have a relatively poor surrounding algebro-geometric theory.

I will describe an older approach, due to Tian and Paul, and some more recent progress enacting Tian's programme (joint work with Rémi Reboulet). This in particular gives a new solution of the Yau-Tian-Donaldson conjecture in the setting of Kähler-Einstein metrics on Fano manifolds. I will also describe some of the algebro-geometric theory of the version of K-stability used in our work, for which we give a geometric (as opposed to numerical) characterisation, and prove various results in the direction of the construction of moduli of varieties.

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Mirko Mauri

Semiampleness in Hodge theory and compact moduli spaces of Calabi--Yau varieties

It is conjectured that every algebraic variety admits a decomposition into varieties with positive, negative, or trivial canonical bundle. While the moduli theory in the cases of positive and negative canonical bundle is by now well understood, the Calabi--Yau case remains substantially more elusive. By the local Torelli theorem, Hodge structures of Calabi--Yau varieties distinguish locally Calabi--Yau varieties. However, due to their inherently transcendental nature, leveraging these structures in the construction of compact algebraic moduli spaces presents significant difficulties. The problem can be reformulated in terms of the semiampleness of the Hodge line bundle associated with certain Calabi--Yau variations of Hodge structure. This property has recently been established in joint work with Benjamin Bakker, Stefano Filipazzi, and Jacob Tsimerman. The result constitutes a key input in the resolution of two long-standing conjectures: Griffiths’ conjecture on functorial compactifications of images of period maps, and the b-semiampleness conjecture of Kawamata, Shokurov and Prokhorov. The proof relies crucially on o-minimality, which provides the perfect framework for taming the transcendence of Hodge theory, and marks the first application of o-minimal methods in birational geometry.

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Junsheng Zhang

Some Applications of Birational Geometry to the Kähler–Ricci Flow

In this mini-course, I will present two results in the study of finite-time singularities of the Kähler–Ricci flow, which make use of techniques from birational geometry. One concerns Kähler–Ricci shrinkers, and the other concerns diameter lower bounds. Part of the material is based on joint work with Song Sun.

Lecture 1. I will introduce the Kähler–Ricci flow and Kähler–Ricci shrinkers, i.e., gradient shrinking Kähler–Ricci solitons. I will discuss their basic properties and survey conjectures and results in the literature.

Lecture 2. I will present joint work with Song Sun showing that every Kähler–Ricci shrinker admits a polarized Fano fibration structure. I will first present the proof in complex dimension two and then in general dimension, using Birkar’s boundedness results in birational geometry.

Lecture 3. Depending on the progress in the first two lectures, time permitting, I will discuss a result on lower bounds for the diameter along finite-time singularities of the Kähler–Ricci flow, based on a weak transcendental base point freeness result.

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RESEARCH TALKS

Simon Jubert

Yau–Tian–Donaldson correspondence for projective bundles over a curve

A central question in complex geometry concerns the existence of canonical metrics. In the 1980s, Calabi proposed extremal metrics as candidates, naturally generalizing Kähler metrics of constant scalar curvature.

In this talk, we will explain that, for projective bundles over a curve, the existence of extremal metrics can be characterized using a notion of stability defined on a certain moment polytope, itself defined in terms of convex functions on this polytope. If times allows, we will also give an interpretation of this notion of stability in terms of test
configurations, that is, one-parameter degenerations of the variety, within the framework of the Yau–Tian–Donaldson conjecture. This is joint work with Chenxi Yin (UQAM).

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Annamaria Ortu

Stability of fibrations and perturbations of the base

On a fibration with smooth fibres, optimal symplectic connections are special metrics which solve a geometric PDE on the fibration. We study the existence of such metrics under perturbations of the Kähler class on the base, and we relate the existence of an optimal symplectic connection to an algebro-geometric stability condition on the fibration. To do this, we employ a perturbation technique based on a moment map interpretation of the optimal symplectic connection equation and the moment map flow. This is joint work (in progress) with Lars Martin Sektnan.

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