Complex geometry seminar

At the beginning of the 20th century, it was known that any compact connected, simply connected Riemann surface is biholomorphic to the projective line.
Subsequently, several characterizations of projective spaces were established. For instance, Siu and Yau stated that projective spaces are the only Kähler manifolds with positive holomorphic bisectional curvature, and Mori proved that they are the only projective manifolds that have an ample tangent bundle. In a different direction, projective spaces are the only Kähler–Einstein manifolds with a positive constant satisfying the equality in the Miyaoka–Yau inequality. This result originating from uniformization theory was generalized in the singular setting by Greb, Kebekus, Peternell and Druel, Guenancia, Păun. More precisely, they characterize singular quotients of $\mathbb{P}^n$ by finite groups acting freely in codimension 1. The aim of this talk is to discuss a generalization of Greb–Kebekus–Peternell’s result in order to characterize quotients of $\mathbb{P}^n$ by any group action.

It is well-known that not every variety in positive characteristic can be lifted to characteristic 0. However, it is conjectured that lifts exist for varieties on which the Frobenius map splits globally—the so-called globally F-split varieties. Recently, Bernasconi, Brivio, Kawakami and Witaszek established the following strong version in two dimension two: globally F-split normal surfaces indeed lift, together with their minimal resolution morphism. From the point of view of the MMP, it is natural to extend this result  to non-normal surfaces that are globally F-split.

In this talk, I will report on a joint project with F. Bernasconi, where we extend this strong lifting statement to non-normal globally F-split CY surfaces. Our argument involves a precise understanding of CY surface pairs with non-empty boundary, and some equivariant MMP.

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 of 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 (pluri)canonical was discovered by Fujita, Kawamata, Kollár and Viehweg. Many fundamental results then followed as a consequence – from moduli theory of such varieties to birational geometry of base spaces 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 ongoing joint work with Sándor Kovács.

Let $X$ be a minimal complex projective variety. Over the past years, many similar inequalities between the Chern classes of $X$ have been obtained. Moreover, it is known precisely which varieties $X$ can achieve the equality. However, so far all results in this direction have focussed on the case where the numerical dimension of $X$ is either very small or very large. In this talk, I will present analogous inequalities for varieties of intermediate Kodaira dimension and I will present a characterisation of those varieties achieving the equality. This talk is partially based on joint work with Masataka Iwai and Shin-ichi Matsumura.

A guiding problem in algebraic geometry is the classification of varieties. In dimension 1, the main invariant for their classification is the genus. Similarly, in higher dimension we study positivity properties of the canonical divisor and a first measure of these is its Iitaka dimension.
A long-standing problem is how we can relate Iitaka dimensions in fibrations: the Iitaka conjectures. Recently, Chang proved an inequality for the Iitaka dimensions of the anticanonical divisors in fibrations over fields of characteristic 0. Both Iitaka’s conjecture and Chang’s theorem are known to fail in positive characteristic. However, in a joint work with Brivio and Chang, we prove that anti-Iitaka holds when the “arithmetic properties” of the anticanonical divisor are sufficiently good.

Géométrie birationnelle des groupes algébriques en caractéristique p>0

(Première partie) Cet exposé portera sur l’étude des familles G ⟶ S de groupes algébriques paramétrées par des variétés algébriques S de caractéristique p>0. Je commencerai l’exposé en expliquant quelques conséquences, pour l’étude des groupes algébriques, de l’existence du morphisme de Frobenius. La géométrie birationnelle est l’étude des différents prolongements possibles d’une famille fixée paramétrée par les points d’un ouvert dense U de S. J’expliquerai la signification de cette étude birationnelle pour la connaissance de toutes les familles. Dans ce contexte, les éclatements de Néron (aussi appelés dilatations) sont l’outil clé pour fabriquer de nouveaux prolongements. Je les présenterai ainsi que quelques développements très récents.
(Deuxième partie) Je me concentrerai ensuite sur le cas des groupes finis et illustrerai les problèmes spécifiques à ce cas. J’introduirai l’espace de modules des prolongements d’une famille fixée, qui est une ind-variété. Enfin j’énoncerai un résultat d’existence de dilatations dans ce cadre.
L’exposé comportera de nombreux exemples.
Il s’agit de résultats obtenus en collaboration avec A. Mayeux et T. RIcharz, ainsi que de travaux d’Alice Bouillet.

The Cremona group of rank N over the field K is the group of birational transformations of the projective N-space that are defined over K. Cremona groups have been studied for a long time but especially the ones of rank 3 and higher remain mysterious, even over the complex numbers. Since 2019, we know that these have non-trivial normal subgroups, due to the construction of quotients by Blanc, Lamy and Zimmermann. In this talk, I will present the following result, obtained in joint work with Blanc and Yasinsky: “Let N be at least 4. Then any group (of cardinality at most the cardinality of the complex numbers) is a quotient of the complex Cremona group of rank N.” The proof uses the Sarkisov program from birational geometry, and Severi-Brauer surfaces from arithmetic geometry.

En 2020, Dey et Kapovich ont montré que le quotient de la boule par un sous-groupe discret et sans-torsion de PU(n,1) est une variété de Stein dès lors que le groupe est convexe-cocompact et que son exposant critique est inférieur à 2. Ils conjecturent que le résultat reste vrai sans l’hypothèse de convexe-cocompacité. Je décrirai des résultats qui impliquent que cette conjecture est vraie pour les groupes géométriquement finis de PU(n,1). J’expliquerai également pourquoi, comme prédit par une autre conjecture de Dey et Kapovich, les seuls quotients de la boule par des sous-groupes convexes-cocompacts d’exposant critique égal à 2 qui ne sont pas des variétés de Stein sont les quotients par un groupe Fuchsien complexe.

Two decades ago, Katzarkov et al. conjectured that a small deformation of a projective variety with big fundamental group still has big $ \pi_1 $. This conjecture was previously known only for surfaces and in some partial cases for threefolds due to Claudon. Recently, in joint work with Mese and Wang, we proved this conjecture when $ \pi_1 $ is linear. In this talk, I will outline the main ideas of the proof and discuss related results on Campana’s broader conjecture concerning the deformation invariance of $ \Gamma $-dimension.