Programme
Lundi
- 9h30-10h30 == Sándor Kovács
- 10h30-11h00 == Pause café
- 11h00-12h00 == Chenyang Xu
- 12h00-14h00 == Lunch
- 14h00-15h00 == Ruadhai Dervan
- 15h00-15h30 == Pause café
- 15h30-16:30 == Jungkai Chen
Mardi
- 9h15-10h15 == Sándor Kovács
- 10h15-10h45 == Pause café
- 10h45-11h45 == Chenyang Xu
- 11h45-14h00 == Lunch
- 14h00-15h00 == Tristan Collins
- 15h00-15h30 == Pause café
- 15h30-16h30 == Behrouz Taji
- 16h30-17h30 == Chung-Ming Pan
- 17h40-20h00 == Réception vin fromage
Mercredi
- 9h15-10h15 == Kenneth Ascher
- 10h15-10h30 == Pause café
- 10h30-11h30 == Kenneth Ascher
- 11h30-14h00 == Lunch
- 14h00-15h00 == Yueqiao Wu
- 15h00-15h30 == Pause café
- 15h30-16h30 == Dori Bejleri
Mini-cours
Kenneth Ascher
Titre: Wall-crossing and explicit moduli spaces of higher dimension varieties
Résumé: Given the constructions of compact moduli spaces of log Fano varieties (K-moduli) and varieties of log general type (KSBA moduli), it is a natural question to understand explicitly the objects parametrized by the boundary of a given moduli problem. Wall-crossing for moduli spaces of pairs, which originated in Hassett’s work on the moduli space of weighted stable curves, has emerged as a powerful technique for studying explicit compactifications of moduli problems. I will review the theory of wall-crossing, and present several applications to explicit moduli problems.
Sándor Kovács
Titre: Moduli theory of higher dimensional varieties of general type
Résumé: In this mini-course I will explain some of the difficulties and their solutions that arise in constructing and even in defining moduli spaces/functors of higher dimensional varieties of general type. The first lecture will concentrate on the moduli theory of KSB stable varieties and the second on that of KSBA stable pairs.
Chenyang Xu
Titre: K-stability and moduli spaces of Fano varieties
Résumé: In the first lecture, I will discuss various equivalent descriptions of K-stability, and explain how to use them to construct K-moduli stacks, parametrizing K-semistable Fano varieties. In the second lecture, I will show K-moduli stacks admit a projective good moduli space, which pointwisely parametrizes K-polystable Fano varieties. I will focus on the most difficult step of establishing the properness, which relies on higher rank finite generations beyond the classical case given by minimal model program and a delicate Langton type argument.
Conférences
Conférencier: Ruadhai Dervan
Titre: Arcs and K-stability
Résumé: K-stability of general polarised varieties is of interest due to its connections to both Kähler geometry and moduli theory (mostly conjectural). Fairly complete results are known for Fano varieties, but comparatively little is known beyond this case. I will discuss a strengthening of K-stability through arcs, and will explain some new results about this stronger version of K-stability.
Arcs are a class of degenerations of a polarised variety, more general than the test configurations involved in the usual definition of K-stability. The main result I will explain gives a new geometric interpretation of this stronger version of K-stability, using the theory of arcs. This geometric interpretation involves orbit-closures of certain pairs of points, as a generalisation of some of the ideas of geometric invariant theory, as introduced and developed by Paul. I will also describe some applications to Kähler geometry, namely a proof of a version of a conjecture of Tian, and a new proof of the Yau-Tian-Donaldson conjecture for Fano manifolds. Broadly speaking, this will mostly be a talk about K-stability beyond the Fano setting, and is all joint work with Rémi Reboulet.
Conférencier: Jungkai Chen
Titre: On threefolds of general type with small volume and genus.
Résumé: It is now known that for threefolds of general type, their volume and genus satisfies the Noether inequality. Following these recent developments, it is known that volume greater or equal to 1 (resp. 2 and 7/2) if genus is 3 (resp. 4 and 5). The canonical models of three-folds with (vol, pg)=(1.3) and (2,4) can be realized as weighted hypersurfaces or weighted complete intersection. In this talk, we are going to introduce the above-mentioned work of Chen-Hu-Jiang and also provide more details about their minimal models. Part of the talk is a joint work in progress with Hsin-Ku Chen.
Conférencier: Yueqiao Wu
Titre: Valuations at infinity
Résumé: The study of valuations centered at a point gives a complete characterization of the metric tangent cone at a klt singularity, which in particular depends purely on the algebraic information. In contrast, as shown in many recent works of complete Calabi—Yau metrics on non-compact Kähler manifolds, this is not the case for metric tangent cones at infinity. Nevertheless, these cones at infinity still arise in degenerations by valuations. We will discuss some joint work in progress with Mattias Jonsson in studying these valuations.
Conférencier: Chung-Ming Pan
Titre: Pluripotential theory in families, stability of coercivity, and singular cscK metrics
Résumé: Searching canonical metrics on compact Kähler manifolds has been a central theme in complex geometry for decades. This talk aims to explain a method for investigating canonical metrics in families of singular varieties, using pluripotential theory and a variational approach in families. We shall start by reviewing the variational picture of constant scalar curvature Kähler (cscK) metrics. I will then introduce a topology on the space of quasi-plurisubharmonic functions in families and explain the variational principle in the family framework. Finally, I will demonstrate the stability of coercivity of the Mabuchi functional and the construction of cscK metrics on smoothable varieties. This talk is based on joint work with T. D. Tô and A. Trusiani.
Conférencier: Behrouz Taji
Titre: Negativity in the direct image of relative anti-canonical bundle in families of Fanos varieties
Résumé: It is well understood that positivity or negativity properties of canonical line bundle encode a significant amount of geometric data about the underlying projective variety. It is therefore unsruprising to expect that the same should be true for the relative canonical divisor for families of projective varieties. For families of varieties whose canonical divisor is ample (canonically polarized) or numerically trivial (Calabi-Yau), important positivity properties of the pushforward of the relative canonical was discovered by Kawamata, Kollár and Viehweg. Many fundamental results then followed as a consequence - from moduli theory of such varieties to birational geometry of their degeneration. For families of Fano varieties however much less is known. In this talk I will discuss how one can complement some of these classical results in the Fano case. This is based on joint work with Sándor Kovács.
Conférencier: Tristan Collins
Titre: Aspects of complete Calabi-Yau metrics on log Calabi-Yau varieties
Résumé: In this talk I will survey some recent results on the existence and structural properties of complete Calabi-Yau metrics on the complement of ample anti-canonical divisors with simple normal crossings. This will discuss joint works with Li, Tong, Yau and Guenancia.
Conférencier: Dori Bejleri
Titre: Proper splittings and valuative criteria for good moduli spaces
Résumé: Given an Artin stack with a good moduli space, the morphism to the good moduli space behaves in many ways like a proper map despite rarely being separated. In this talk, I will explain two results in this direction. The first is the existence of generically finite proper coverings of the stack by a scheme, generalizing the fact that separated Deligne-Mumford stacks admit finite covers by schemes. The second is a strong version of the existence part of the valuative criterion of properness for good moduli space morphisms which generalizes a result of Bresciani-Vistoli for tame stacks. Applications include compactifications of families and projectivity criteria for good moduli spaces. This is based on joint work with Elmanto and Satriano and joint work in progress with Inchiostro and Satriano respectively.