MARGE in Brest
Fibrations and Deformations

June 17

09:00 Rémi Delloque

Title: Perturbations of vector bundles whose curvature solves a polynomial equation

Abstract: Start from a complex Hermitian holomorphic vector bundle and assume that its curvature form satisfies a polynomial equality whose coefficients are closed forms, like the Hermitian Yang--Mills equation for example. When we perturb the coefficients of the equation and the complex structure of the bundle, under which conditions can we find a Chern connection close to the initial one in a given gauge orbit whose curvature form is a solution to the new equation ? Under suitable hypothesis on the initial complex structure, a local Kobayashi-Hitchin like correspondence is shown by using a moment map interpretation and geometric invariant theory techniques. If time allows it, I will also expose continuity results about the solutions of the equation in function of the coefficients and the gauge orbit where we look for a solution. This works registers in my PhD thesis under the direction of Carl Tipler.

11:00 Shing Tak Lam

Title: Canonical metrics on families of vector bundles

Abstract: A main goal of complex geometry is to find canonical metrics on an object, such as cscK metrics on manifolds or Hermite-Einstein metrics on vector bundles. Dervan-Sektnan and Ortu have introduced a version of this story for fibrations, which should be thought of as families of manifolds. I will describe a new theory of canonical metrics for families of Hermite-Einstein vector bundles, or more generally, families of slope semistable vector bundles. Concretely, this produces a new PDE for a Hermitian metric on a family of bundles, which I conjecture satisfies many similar properties to the Hermite-Einstein equation, such as connections with algebro-geometric stability and moduli theory. The main result is a new construction of Hermite-Einstein metrics in adiabatic limits over products of manifolds, in the spirit of Dostoglou-Salamon, Hong, Fine, Sektnan-Tipler et al., assuming this new PDE is solvable.

14:00 Lars Sektnan

Title: CscK metrics in perturbed Kähler classes

Abstract: A classical theorem of LeBrun-Simanca says that the set of Kähler classes admitting a cscK metric on a Kähler manifold is open if the manifold has discrete automorphism group. More generally the same openness result holds if one instead considers extremal metrics. Openness does not hold in general, however, when one perturbs the complex structure instead of the Kähler class. Ortu recently showed that for a cscK manifold with automorphisms, K-polystability is exactly what is needed to ensure that one can obtain a cscK metric for a nearby complex structure. Building on this technique, I will consider a couple of contexts where we vary both the complex structure and the Kähler class. This is joint work with Carl Tipler and work in progress with Annamaria Ortu.

June 18

09:00 Ruadhai Dervan

Title: Scalar curvature as a moment map, with applications

Abstract: Donaldson and Fujiki's moment map interpretation of the scalar curvature has been extremely influential in our understanding of the geometry behind scalar curvature. I will begin by describing a new approach to scalar curvature as a moment map, allowing variation of both complex and Kähler structure (in fact applying to general equivariant fibrations), which is in some sense closer to algebraic geometry. The new moment map interpretation is useful for deformation problems, and I will also describe a general solution to the Arezzo-Pacard problem, around the existence of extremal Kähler metrics on blowups, which uses this moment map interpretation as a key ingredient. The moment map part of the talk is joint work with Michael Hallam, while the Arezzo-Pacard part is joint work with Lars Sektnan.

11:00 Pietro Piccione

Title: A non-Archimedean approach to the Yau-Tian-Donaldson Conjecture

Abstract: In Kähler Geometry, the Yau-Tian-Donaldson Conjecture relates the differential geometry of compact Kähler manifold with an algebro-geometric notion called K-stability. I will start with a brief overview of the topic, and then I will discuss a possible non-Archimedean approach to solve this conjecture, generalizing a result of Chi Li to the transcendental setting.

14:00 Tran Trung Nghiem

Title: Complete Calabi-Yau metrics on affine smoothings of irregular toric cones

Abstract: I will present a joint work with Ronan J. Conlon where we exhibit infinitely many new examples of affine Calabi-Yau manifolds of Euclidean volume growth and quadratic curvature decay with irregular Calabi-Yau asymptotic cones. Metrics of this kind are surprisingly rare in the literature, given that those with toric smooth base asymptotic cones can only arise in dimension 3. After a brief survey on the state of the art, I will consider an explicit family of three-dimensional Gorenstein toric cones and show that each cone has a non-trivial affine smoothing via Altmann's deformation theory of toric varieties. Irregularity of the Calabi-Yau metric is derived by an explicit computation of the normalized volume and the Reeb minimizer via Martelli-Sparks-Yau localization formula. Existence of the Calabi-Yau metrics on the affine smoothings with desired property then follows from Conlon-Hein's classification theory.

June 19

09:00 Simon Jubert

Title: Weighted cscK problem on semisimple principal fibrations

Abstract: I will present some recent progress on canonical metrics on semisimple principal fibrations. In particular, I will discuss how the existence of a weighted constant scalar curvature Kähler metric on the total space can be characterized in terms of the coercivity of a weighted Mabuchi energy on the fiber. This extends some results of my PhD beyond the extremal case. In the second part, we will focus on weighted solitons, and I will introduce a numerical invariant that characterizes this coercivity condition. This work is based on a joint collaboration with Di Nezza--Lahdili, and Delcroix.