Résumés des exposés

Gilles Carron: Kato conditions : examples

This is part of a joint work with Ilaria Mondello (Creteil) and David Tewodrose (Bruxelles)

I will explain some geometric/analytic condition insuring a Kato bound on the Ricci curvature. And I will describe a non collapsed strong Kato limit that is branching.

Thibaut Delcroix: Weighted analytic delta and greatest weighted Ricci lower bound

I will report on joint work (in progress) with Simon Jubert on the weighted analytic delta invariant and the greatest weighted Ricci lower bound for compact Kähler manifolds. Among our results, I will discuss some conditions of existence of canonical Kähler metrics as well as a lower bound on the (algebraic) delta invariant of a Fano semisimple principal fibration whose basis is not assumed cscK, in terms of its basis and its (weighted) fiber.

Alix Deleporte: Semiclassical analytic methods for the Bergman kernel and applications to the quantization of Mabuchi geodesics

In the last few years, decisive progress has been made in the understanding of the Bergman kernel and associated quantum objects on a Kähler manifold with real-analytic metric. More precisely, one now understands the semiclassical Bergman kernel on a polarised Kähler manifold up to an error O(exp(-ck)), where k→∞ is the power of the tensor bundle.

In this talk, I will present an overview of these results and the underlying techniques. As an application, I will describe recent work in collaboration with S. Zelditch (Northwestern) on the quantization of the initial value problem for the geodesics on the space of Kähler structures.

Alix Deruelle: Ancient solutions coming out of 4-dimensional spherical orbifolds

Given a 4-dimensional Einstein orbifold that cannot be desingularized by smooth Einstein metrics, we investigate the existence of an ancient solution to the Ricci flow coming out of such a singular space. In this talk, we will focus on singularities modeled on a cone over RP3 that are desingularized by gluing Eguchi-Hanson metrics to get a first approximation of the flow. We show that a parabolic version of the corresponding obstructed gluing problem has a smooth solution: the bubbles are shown to grow exponentially in time, a phenomenon that is intimately connected to the instability of such orbifolds.

Shubham Dwivedi: Flows of Spin(7)-structures

We will discuss the negative gradient flow of an energy functional of Spin(7)-structures on compact 8-manifolds. The energy functional is the L2-norm of the torsion of the Spin(7)-structure. We will talk about the short-time existence and uniqueness of solutions to the flow. We will also explain how this negative gradient flow is the most general flow of Spin(7)-structures.

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 polarized fibration is associated with a 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. If time permits, we discuss a relation of the above results with the conjecture on the optimality of Holomorphic Morse Inequalities due to Demailly.

Henri Guenancia: Degeneration of conic Kähler-Einstein metrics

I will report on a joint work with Olivier Biquard. We show that given a Fano manifold X and a smooth divisor D satisfying suitable conditions, one can construct positively curved Kähler-Einstein metrics with cone singularities along D which, after rescaling, converge to the complete Ricci flat Tian-Yau metric on X\D when the cone angle approaches its critical value.

Jakob Hultgren: Real Monge-Ampère equations and the SYZ conjecture for hypersurfaces in toric Fano

It is well known since a few years that a weak metric version of the SYZ conjecture follows from existence and/or structural results about Monge-Ampère equations. For general maximal degenerations of Calabi-Yau manifolds, this principle is formulated in terms of non-Archimedean geometry, but for certain families of hypersurfaces in toric Fano manifolds, the weak metric SYZ conjecture follows from solvability of a real Monge-Ampère equation on the boundary of a polytope. I will present positive and negative existence results in this setting and then talk about the question of regularity of solutions to this equation, and what kind of geometric information is contained in the answer to this question. This is based on joint work with Rolf Andreasson, Thibaut Delcroix, Mattias Jonsson, Enrica Mazzon and Nick McCleerey.

Simon Jubert: Coercivity of the weighted Mabuchi functional and weighted cscK metrics

In this talk, we will discuss the existence of a large class of Kahler metrics with a special curvature condition, called weighted cscK metric. These metrics were introduced by Lahdili with the goal of unifying several natural problems in Kahler geometry such as finding cscK metrics, Kahler-Ricci solitons (and their weighted extensions), extremal metrics on holomorphic fibrations, and others. I will present recent work in collaboration with Lahdili and Di Nezza, where we demonstrate that the existence of a weighted cscK metric is equivalent to the coercivity of the weighted Mabuchi functional. First, I will introduce the notion of weighted cscK metric. Next, I will discuss applications of the result. Finally, I will explain the proof strategy, which is a generalization of that used by Chen and Cheng in the cscK case.

Éveline Legendre: From Kähler Ricci solitons to Calabi-Yau Kähler cones

In this talk, I will explain a recent result obtained in collaboration with V.Apostolov and A. Lahdili (UQAM, Canada) where we show that the cone over the product of a smooth Fano Kähler Ricci soliton with a complex projective space of sufficiently large dimension is a Calabi Yau cone. This can be seen as an asymptotic version of a conjecture by Mabuchi and Nikagawa.

Jason Lotay: Joyce conjectures for Lagrangian mean curvature flow of surfaces

Lagrangian mean curvature flow is a potentially powerful tool for tackling problems related to symplectic topology. Joyce made a series of conjectures about the singularity formation, surgeries and long-time behaviour in Lagrangian mean curvature flow. I will describe recent progress on these conjectures in the setting of 4-manifolds, where the Lagrangians are surfaces.

Heather Macbeth: Sharp L^2 estimates for the drift heat equation on Ricci solitons

Several classical estimates can be (re)-interpreted as providing L^2 estimates, with time-dependent Gaussian weights, for the drift heat equation on a (complete gradient shrinking) Ricci soliton. I will describe these, and then present a new such estimate, which is sharp. Finally, I will describe a second motivation for the sharp estimate: when transferred and scaled to an estimate for the heat equation along the Ricci flow of the soliton, this estimate is uniform up to the singular time.

Ilaria Mondello: Limits of manifolds with a Kato bound on the Ricci curvature

In this talk I will present a joint work with Gilles Carron (Nantes) and David Tewodrose (Bruxelles). The goal is to present how an integral condition on the Ricci curvature, related to the heat kernel and inspired by Kato potentials in R^n, allows us obtain many regularity results for limit spaces that recover Cheeger-Colding theory.

Léonard Pille-Schneider: The SYZ conjecture for families of 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.

Rémi Réboulet: Arcs and the Hilbert-Mumford criterion for K-stability

A conjecture of Tian asserts that coercivity of the Mabuchi functional of a polarised manifold (X,L) (which detects cscK metrics) on Fubini-Study metrics is equivalent to K-stability with respect to one-parameter subgroups acting on X embedded into projective space by a very ample exponent of L. Recent work of Paul-Sun-Zhang together with an older result of Paul show that this conjecture cannot hold as stated. I will talk about work joint with Ruadhaí Dervan where we prove an optimal version of Tian's conjecture, relying on arcs instead of one-parameter subgroups, as were introduced by Donaldson and Wang. If time permits, I will also explain applications regarding the automorphism group of (X,L).

James Stanfield: Homogeneous generalized Ricci flows

The generalized Ricci flow, or GRF (first studied by Callan-Friedan-Martine-Perry, and later by Streets-Garcia-Fernandez) is an analogue of the Ricci flow in the setting of Hitchin's generalized geometry. It is a certain super-solution of Ricci flow coupled with the heat flow on 3-forms and subsumes several geometric flows, including the Ricci flow, the generalized Kähler-Ricci flow, and the pluriclosed flow. The latter is a flow of pluriclosed (strong Kähler with torsion, or SKT) metrics on Hermitian manifolds, introduced by Streets and Tian. The GRF also appears in the physics literature as a renormalization group flow (see, e.g., Oliynyk, Suneeta, and Woolgar (2006)).

In this talk, we discuss the behavior of the GRF on (discrete quotients of) Lie groups. We establish the global existence of the flow on solvmanifolds in arbitrary dimensions—a result that is new even for the pluriclosed flow. We also define a notion of generalized Ricci soliton that allows for non-trivial expanding examples. On nilmanifolds, we show that these solitons arise as rescaled limits of the GRF under certain circumstances. Our main tool is an adaptation of Lauret's "bracket flow" to the GRF, and a new formula for the generalized Ricci curvature in terms of the moment map for the action of a real-reductive Lie group on the space of generalized Lie brackets. This is based on joint work with Elia Fusi (Università di Parma) and Ramiro Lafuente (The University of Queensland).

Antonio Trusiani: Extremal Kähler metrics on modifications

I will present the invariance of extremal Kähler manifolds under a suitable class of bimeromorphic morhpisms. This is a joint work with M. Jonsson and S. Boucksom, and it generalizes previous results of Arezzo-Pacard-Singer, Seyyedali-Székelyhidi and Hallam.

I will show how our main result is obtained as a consequence of a general uniform coercivity estimate for the Mabuchi energy on the modification, which applies more generally to the class of weighted extremal metrics, modulo a log-concavity assumption on the first weight, and to any equivariant resolution of singularities of Fano type of a compact Kähler klt space whose weighted Mabuchi energy is assumed to be coercive.

David Witt Nyström: Competitive Hele-Shaw flows and quadratic differentials

This talk is based on joint work with Fredrik Viklund. In the classical Hele-Shaw flow a domain in the complex plane grows according to the gradient of its Green's function, thus modelling the propagation of a viscous fluid trapped in a thin layer. We introduce a competitive version of the flow where several domains in the complex plane (or more generally in a Riemann surface of finite type) similarly strive to expand but at the same time hinder each other. Interestingly, stationary flows correspond to a special class of quadratic differentials whose associated half-translation surfaces have a simple description. We also introduce a discrete model, closely related to Propp's competitive erosion model, which conjecturally allows us to simulate the flow.

Michela Zedda: Projectively induced and balanced metrics on noncompact Kähler manifolds

The aim of the talk is to give an overview of the problems related to the existence of Kähler-Einstein and constant scalar curvature Kähler metrics induced by a holomorphic immersion into the complex projective space, focusing in particular to the case when the manifold is noncompact and the immersion is balanced. The talk is based on joint work with Simone Cristofori (Univ. of Parma), Andrea Loi and Fabio Zuddas (Univ. of Cagliari).

Shengxuan Zhou: Examples related to Ricci limit spaces and topology

In this talk, we will describe the construction of two examples related to Ricci limit spaces:

(1) For any n≥3, there exists an n-dimensional Ricci limit space that has no open subset which is topologically a manifold. This generalizes a result of Hupp-Naber-Wang. As a corollary, our example provides a collapsed sequence of boundary free manifolds whose limit has a dense boundary with infinitely many connected components.

(2) For any n≥4, there exists a sequence of n-dimensional tori with Ricci lower bound that converges to a singular space. This answers a question posted by Petrunin and Brue-Naber-Semola. In the 4-dimensional case, we prove that the Gromov-Hausdorff limit of tori with a two-sided Ricci bound and a diameter bound is always a topological torus.

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