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Basel-Dijon-EPFL seminar
Basel, November 29-30, 2021
Schedule
In Vesalianum - Kleiner Hörsaal (O1.13), Vesalgasse 1, 4051 Basel
Monday
November 29
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Tuesday
November 30
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9h15-10h15
Benoît Cadorel
Complex hyperbolicity, VHS and Higgs bundles
10h45-11h45
Andriy Regeta
When is the automorphism group of an affine variety nested?
12h00-13h00
Pierre-Marie Poloni
Real forms of affine algebraic varieties: two examples
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| lunch |
lunch |
14h00-15h00
Alan Muniz
Symmetries of foliations on surfaces
15h30-16h30
Sokratis Zikas
Rigid birational involutions of ℙ3
17h00-18h00
Fabio Bernasconi
On the log liftability of F-split surfaces
social dinner
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Fabio Bernasconi (EPFL) - On the log liftability of F-split surfaces
Given a projective variety X over an algebraically closed field of characteristic p>0, it is an interesting question to understand
the possible geometric and arithmetic obstruction for the existence of a lifting to characteristic zero.
In this direction, motivated
by the case of abelian varieties and K3 surfaces, it is
conjectured that ordinary
Calabi-Yau varieties
should admit a lifting to characteristic zero.
In this talk I will report a joint work with I. Brivio, T. Kawakami
and J. Witaszek where we show that globally F-split
surfaces (which can be thought as arithmetically well-behaved log Calabi-Yau pairs) are log liftable to characteristic zero.
As a corollary we deduce the Bogomolov bound on the number of
singular points of F-split klt del Pezzo surfaces (which was known to be false without the F-splitness
condition).
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Benoît Cadorel (Nancy) - Complex hyperbolicity, VHS and Higgs bundles
Given a complex projective manifold of general type X, it is in general a very difficult problem to determine the locus of its entire curves, that is, the images of holomorphic maps from the complex plane with values in X. If this locus is not Zariski dense, we say that X is quasi-hyperbolic. The Green-Griffiths-Lang conjecture then states that for projective manifolds, this property should be equivalent to being of general type; this remains a widely open question in general.
The previous conjecture has been a guiding question for many moduli problems for which the parameter spaces is known to have negative curvature properties. This has lead to an important amount of work in the last few years (starting with Griffiths, Schmid, Viehweg, Zuo... and more recently with Brunebarbe, Rousseau, Deng, Brotbek,...), and has permitted to obtain strong hyperbolicity results for many interesting classes of varieties, in particular the ones supporting variations of Hodge structures (VHS). In this talk, I will present a recent work with Y. Deng, where we prove hyperbolicity properties of varieties supporting a nilpotent harmonic Higgs bundle, which form a natural generalization of VHS.
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Alan Muniz (Dijon) - Symmetries of foliations on surfaces
Let X be a smooth complex projective surface. A foliation F on X is given by a subsheaf of the tangent sheaf, TF -> TX. Under some positivity condition on TF* we have finiteness for the group Aut(F) of automorphisms of X that preserve F. In this talk we will address the problem of bounding the order of Aut(F).
Under mild conditions on X and F we will show that |Aut(F)| is bounded above by a degree 3 polynomial on the Chern numbers of F and X. This is done by analysing invariant loci for subgroups of Aut(F). We will also prove, via a slightly different approach, a linear bound for foliations on the projective plane X = P2. This talk is based on joint works with Mauricio Corrêa and Rudy Rosas.
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Pierre-Marie Poloni (Basel) - Real forms of affine algebraic varieties: two examples
We will show that the famous Russell cubic threefold admits a unique isomorphism class of real forms and that all surfaces of equation xy=p(z) (often called Danielewski surfaces) admit at most six isomorphism classes of real forms.
This is joint work with Jérémy Blanc and Anna Bot
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Andriy Regeta (Jena) - When is the automorphism group of an affine variety nested?
In this talk we will discuss the structure of the automorphism group of an affine variety.
More precisely, we conjecture that the following conditions on the neutral component of the automorphism group are equivalent:
(1) it is equal to the union of all algebraic subgroups;
(2) it is exhausted by an inductive limit of algebraic subgroups (then we call it nested);
(3) it is a semidirect product of an algebraic torus and a direct limit of abelian unipotent algebraic groups;
(4) its unipotent elements comprise an abelian subgroup.
This conjecture holds in dimension 2 and partially in arbitrary dimension: the equivalences hold for a subgroup of the automorphism group generated by connected algebraic subgroups.
This talk is based on the joint work with Alexander Perepechko.
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Sokratis Zikas (Basel) - Rigid birational involutions of ℙ3
The Cremona group is the group of birational transformations of the projective n-space over some field k.
The study of these groups dates back to the 19th century with some of the central questions still being open.
In the recent years new techniques, based on the Minimal Model Program, have been developed to answer some of these questions over the field of complex numbers.
In this talk, using these techniques, I will explain how to construct families of birational involutions on the projective 3-space which do not fit in an elementary relation of Sarkisov links.
Using these involutions, we can construct new homomorphisms from the Cremona group, effectively reproving non-simplicity, and show that it admits a free product structure.
Finally, using the free product structure, we will show that the group of automorphisms of the Cremona group is not generated by inner and field automorphisms.
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Participants
Ahmed Abouelsaad (Basel)
Jefferson Baudin (EPFL)
Vladimiro Benedetti (Dijon)
Fabio Bernasconi (EPFL)
Jérémy Blanc (Basel)
Anna Bot (Basel)
Benoît Cadorel (Nancy)
Jung-Kyu Canci (Basel)
Adrien Dubouloz (Dijon)
Mani Esna Ashari (Basel)
Daniele Faenzi (Dijon)
Stefano Filipazzi (EPFL)
Pascal Fong (Basel)
Pierre-Alexandre Gillard (Dijon)
Tanuj Gomez (Freiburg)
Hanspeter Kraft (Basel)
Philipp Mekler (Basel)
Alan Muniz (Dijon)
Irène Meunier (Basel)
Pierre-Marie Poloni (Basel)
Quentin Posva (EPFL)
Andriy Regeta (Jena)
Javier Carvajal Rojas (EPFL)
Roberto Svaldi (EPFL)
Ronan Terpereau (Dijon)
Christian Urech (EPFL)
Immanuel Van Santen (Basel)
Sokratis Zikas (Basel)
Send an email to Jeremy  Blanc  unibas ch if you would like to participate.
Organisers
Jérémy Blanc (Basel)
Adrien Dubouloz (Dijon)
Christian Urech (EPFL)
Roberto Svaldi (EPFL)
Ronan Terpereau (Dijon)
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