La semaine de l’IMB

Dilation surfaces are surfaces modeled after the complex plane whose structure group is generated by the group of translations and dilations. Given a dilation surface, for any direction in $S^1$ there exists a corresponding directional foliation on the surface. In this talk, we will study the four possible types of dynamical behaviour that such a foliation may have (i.e completely periodic, Morse-Smale, minimal or Cantor-like) and deduce a dynamical decomposition theorem for the directional foliation on dilation surfaces using results of C.J. Gardiner and G. Levitt from the 1980s.

In a second step, we study the first return map of the directional foliation on a dilation surface, which is a so-called affine interval exchange transformation (AIET). We introduce a powerful tool called Rauzy-Veech induction in order to develop a renormalization scheme which allows to find a decomposition of any given AIET into finite union of intervals which exhibit only one of the four types of dynamical behaviour. This provides an alternative, purely combinatorial approach to the decomposition results of Levitt and Gardiner and is joint work with Corinna Ulcigrai and Charles Fougeron.

Cet exposé concerne un travail en collaboration avec Tien-Cuong Dinh, Hsueh-Yung Lin, Keiji Oguiso, Long Wang et Xun Yu. Soit X une variété algébrique complexe. Les formes réelles de X sont les variétés réelles W dont la “complexification”, en tant que variété complexe, est isomorphe à X. Bien entendu, certaines variétés complexes n’ont pas de forme réelle. Un fait plus surprenant, mis en évidence par Lesieutre en 2016, est l’existence d’une variété complexe admettant une infinité de formes réelles. Dans cet exposé, on présente une surface de rang de Picard relativement petit possédant une infinité de formes réelles. L’exemple en question est obtenu en adaptant une construction de Dinh-Oguiso-Yu à base de surfaces K3 via une technique due à Mukai. En fin de compte, on fabrique une surface d’Enriques dont l’éclatement en un point très général d'une courbe bien choisie possède une infinité de formes réelles. Si le temps le permet, on expliquera aussi pourquoi le groupe d’automorphismes de cet éclatement n’est pas de type fini.

We take a section P of infinite order on an elliptic surface and consider points where some multiple nP is tangent to the zero section (These are "unlikely intersections" and our consideration of them is motivated by a question in geography of surfaces. It is also analogous to the question of whether elements of an elliptic divisibility sequence are square-free.) In characteristic zero, we show finiteness and give a sharp upper bound, relying heavily on a canonical parallel transport in a family of elliptic curves (the "Betti foliation") and a certain real-analytic one-form. Although the finiteness statement looks completely reasonable in characteristic p, it's not clear what would replace the (non-algebraic) 1-form. Time permitting, I will explain how ongoing work with Felipe Voloch connects tangencies to the p-descent map and allows us to bound them in characteristic p as well.

(w/ G. Urzua and F. Voloch)