Serge Cantat (Rennes), Maximal subgroups and the Cremona group
Abstract:
Using ideas from the proof of Tits’ alternative, Margulis and Soifer proved that a finitely generated group of matrices which is not virtually solvable contains uncountably many maximal subgroups; in particular, it contains maximal subgroups of infinite index. I will describe this theorem and explain how the proof can be adapted to other contexts, for example to the Cremona group in two variables.
Davoud Cheraghi (Imperial), Rotation domains, nearly rotation domains, and their global biholomorphic geometry
Abstract:
A rotation domain (Siegel disk) of a holomorphic map on a complex manifold is an invariant domain on which the map is biholomrphically conjugate to a rotation. While there are strong local results about their existence, there is little known about the global complex geometry of rotation domains. We present a brief survey of results in one and higher dimensions, including invariant nearly-rotation-domains, and present a new method to build automorphisms of C^2 with round rotation domains.
Charles Favre (Paris), Automorphisms and pseudo-automorphisms
Abstract:
Joint work with Alexandra Kuznetsova
Whereas automorphisms on projective surfaces with positive entropy are somehow abundant, it is much harder to find such maps in higher dimensions. We shall discuss the case of birational maps defined on a family of polarized abelian varieties.
Stefano Filipazzi (EPFL), On semi-ampleness of the moduli part
Abstract:
Kodaira's study of elliptic surfaces shows that one can use the j-function of elliptic curves to study the variation in moduli of the fibres of an elliptic surface f:S->C, thus defining a "moduli divisor". Via Hodge theory, this perspective is known to generalise to fibrations whose fibres are K-trivial varieties. Recent work of Ambro, Cascini, Shokurov, and Spicer shows that methods from foliations apply to introduce a moduli divisor for general fibrations. In this talk, I will discuss positivity properties of the moduli divisor of a general fibration. This is joint work with Spicer.
Cecile Gachet (Nice), The cone conjecture for certain Calabi-Yau manifolds obtained as fiber products
Abstract:
The cone conjecture is a long-lasting conjecture in birational geometry, that can be viewed as a hypothetical counterpart to the cone theorem. While the cone theorem describes the K-negative part of the nef cone, the cone conjecture aims at describing the nef cone of K-trivial varieties, or pairs. It predicts for any K-trivial variety X the existence of a rational polyhedral fundamental domain for the action of the group Aut(X) on the cone Nef^e(X). If the conjecture is true for a certain X, it has important consequences, such as the finiteness of real forms of X.
In this talk, after discussing the relationship between cone conjecture and finiteness of real forms, I present a construction inspired by the work of Schoen, Namikawa and Grassi-Morrison, of Calabi-Yau manifolds obtained as fiber products over the projective line. If time allows, I intend to sketch the proof of the the cone conjecture for these Calabi-Yau manifolds. This is joint work with Hsueh-Yung Lin and Long Wang.
Julia Schneider (EPFL), Birational maps of Severi-Brauer surfaces
Abstract:
Cremona groups are groups of birational transformations of a projective space. Their structure depends on the dimension and the field. In this talk, however, we will focus on birational transformations of (non-trivial) Severi-Brauer surfaces, that is, surfaces that become isomorphic to the projective plane over the algebraic closure of K. Such surfaces do not contain any K-rational point. We will prove that if such a surface contains a point of degree 6, then its group of birational transformations is not generated by elements of finite order as it admits a surjective group homomorphism to the integers.
As an application, which will be discussed in the follow-up talk by J. Blanc, we use this result to study Mori fiber spaces over the field of complex numbers, for which the generic fiber is a non-trivial Severi-Brauer surface. We prove that any group of cardinality at most the one of the complex numbers is a quotient of the Cremona group of rank 4 (and higher).
This is joint work with Jérémy Blanc and Egor Yasinsky.
Robert Svaldi (Milan), Birational geometry of surface foliations: towards a moduli theory
Abstract:
The birational classification of foliated surface is pretty much complete, thanks to the work of Brunella, Mendes, McQuillan.
The next obvious step in this endeavour, in analogy with the classical case of projective varieties and log pairs, is to construct moduli spaces for foliated varieties (starting from the general type case).
The first question to ask, on the road towards constructing such a moduli space, is how to show that foliated varieties of fixed Kodaira dimension are bounded, that is, they come in finitely many algebraic families — provided, of course, that we fix certain appropriate numerical invariants. It turns out that, to best answer this question, rather than working with the canonical divisor of a foliation it is better to consider linear systems of the form $|nK_X + mK_{\mathcal F}|$, $n,m >0$, as those encode a lot of the positivity features that classically the canonical divisor of an projective variety displays.
In this talk, I will introduce this framework and explain how this approach leads to answering the question about boundedness for foliated surfaces. Time permitting, I will address also what
This talk features joint work with C. Spicer, work with J. Pereira, and work in progress with M. McQuillan and C. Spicer.