Monday June 19th
14h-15h: Ilia Itenberg (Sorbonne Université)
15h-15h30: coffee break
15h30-16h30: Guillaume Rond (Université Aix Marseille)
Tuesday June 20th
10h-11h : Mario Kummer (Technische Universität Dresden)
11h-11h30: coffee break
11h30-12h30: Jules Chenal (Universtié de Lyon 1)
14h-15h: Diego Matessi (Universita' degli Studi di Milano)
15h-15h30: coffee break
15h30-16h30: Patrick Popescu-Pampu (Université de Lille)
Wednesday June 21st
10h-11h : Antoine Toussaint (Sorbonne Université)
11h-11h30: coffee break
11h30-12h30: Arthur Renaudineau (Université de Lille)
14h-15h: François Bernard (Université d'Angers)
15h-15h30: coffee break
15h30-16h30: Santiago Toro Oquendo (Université de Brest)
TITLE and ABSTRACT
François Bernard (Université d'Angers) : Locally Lipschitz rational functions on complex algebraic varieties.
Inspired by the work of Pham and Teissier about locally Lipschitz meromorphic functions on complex analytic varieties, we use the recent researches on complex regulous functions to study locally Lipschitz rational functions on complex algebraic varieties. As in the analytic case, we prove that locally Lipschitz rational functions correspond to the regular functions on an algebraic variety called the "Lipschitz saturation" which is obtained by an algebraic process. We deduce from this result conditions for two complex algebraic varieties to be birationally and locally bilipschitz equivalent.
Jules Chenal (Universtié de Lyon 1) : Symmetry and Degeneracy of the Kalinin Spectral Sequence of Renaudineau and Shaw,
In Bounding the Betti Numbers of Real Hypersurfaces near the Tropical Limit, A. Renaudineau and K. Shaw presented a filtration of the cellular complex of a real, primitively patchworked, hypersurface in a non singular toric variety. This filtration, inspired by the work of Kalinin, allows, by computation of its first page, for individual bounds on the Betti of such hypersurface. We would like to present two results on this spectral sequence. The first is the symmetry of every page following the first. The second is a degeneracy criterion. It reformulates the vanishing of the boundaries of the spectral sequence in terms of the homological inclusion of the hypersurface in its surrounding toric variety. As an example we use this criterion to show that the spectral sequence degenerates at the second page in some cases.
Ilia Itenberg (Sorbonne Université) : Empty real plane sextic curves
Many geometric questions concerning K3-surfaces can be restated and solved in purely arithmetical terms, by means of an appropriately defined homological type. For example, this works well in the study of singular complex sextic curves or quartic surfaces, as well as in that of smooth real ones.
However, when the two are combined (singular real curves or surfaces), the approach fails as the natural concept of homological type does not fully reflect the geometry. We show that the situation can be repaired if the curves in question have empty real part;
then, one can confine oneself to the homological types consisting of the exceptional divisors, polarization, and real structure. The resulting arithmetical problem can be solved, and this leads to an equivariant equisingular deformation classification of real plane sextics with empty real part.
This is a joint work with Alex Degtyarev.
Mario Kummer (Technische Universität Dresden): Ulrich sheaves and degrees of morphisms
We explain a connection between Ulrich sheaves and topological properties of the real part of a projective variety. For instance, if an embedded projective variety is the support of a certain Ulrich sheaf, then the degree of all its linear projections (restricted to the real part) can be read off in an explicit way from the free resolution of the Ulrich sheaf. These results can be generalized to the context of A^1-enumerative geometry. As an application, we define an arithmetic version of Viro's encomplexed writhe for curves in projective space over an arbitrary field which can be considered to be an arithmetic analogue of a link invariant. This is joint work with Daniele Agostini.
Diego Matessi (Universita' degli Studi di Milano) : Mirror symmetry and real structures
I will speak about joint work with A. Renaudineau. First I will review two approaches to mirror symmetry of Calabi-Yau manifolds. The first one (following Batyrev) considers pairs of dual reflexive polytopes and the anticanonical hypersurfaces inside the associated toric varieties. The second one (following the SYZ conjecture) considers dual Lagrangian torus fibrations. In both pictures it is possible to incorporate the data of a real structure: patchworking with repsect to central subdivisions in one case, fibre-preserving involutions in the other. These real structures are parametrized by mod 2 divisor classes in the mirror. I will then present some results and conjectures which relate the cohomology of the real part to the cup product of divisor classes in the mirror.
Patrick Popescu-Pampu (Université de Lille) : Combinatoire de morsifications analytiques réelles
Je présenterai un travail fait en collaboration avec A. Bodin, E. García Barroso et M.-S. Sorea. Nous étudions une large classe de morsifications de germes de fonctions analytiques réelles à une variable. Nous caractérisons le type combinatoire des fonctions de Morse associées en termes d'un arbre de contact construit à partir des racines de Newton-Puiseux réelles de la courbe polaire de la morsification.
Arthur Renaudineau (Université de Lille) : Relations between Welschinger invariants of the ellipsoid and of RP^3.
After a suggestion of Kollár, Brugallé and Georgieva proved that Welschinger invariants of RP^3 can be computed in terms of Welschinger invariants of the hyperboloid. In a joint work in progress with Brugallé, we prove that one can express the Welschinger invariants of the ellipsoid in terms of Welschinger invariants of RP^3 in the case that there is at least 2 real points in the configuration, and simply in terms of the Gromov-Witten invariants of RP^3 when there is only 1 real point in the configuration. The proof goes by considering so called relative Welschinger invariants and proving a recursive equation for those invariants.
Guillaume Rond (Université Aix Marseille) : Algébrisations et déformations équisingulières.
Après avoir donné les définitions nécessaires et présenté le contexte, j'expliquerai comment on peut obtenir un ensemble algébrique Y défini par des équations à coefficients algébriques qui soit homéomorphe à un ensemble algébrique X donné. Les équations définissant cet ensemble algébrique Y sont obtenues à partir de celles définissant X en perturbant correctement leurs coefficients. C'est un travail en commun avec Adam Parusinski.
Santiago Toro Oquendo (Université d'Angers) : C_2-Isovariant simplicial homotopy in real algebraic geometry.
Classically equivariant algebraic topology focuses on studying and defining algebraic invariants of topological spaces with group actions. Voevodsky's and Morel's work have provided new insights on how to use many techniques of algebraic topology in algebraic geometry. Recently Mark Behrens and Jay Shah provided methods to compute C_2 equivariant homotopy groups of the Betti realization of certain motivic spectra over R. In this talk, I will discuss how to construct an isovariant homotopy theory for spaces carrying an action of the C_2-group by means of Cisinski's techniques on model structures for presheaf categories. This theory has been constructed to study real Betti realization functors with values in the isovariant homotopy category. Part of this work (some still in progress) is a joint work with Johannes Huisman.
Antoine Toussaint (Sorbonne Université) : Real Structures of Phase Tropical Surfaces
Phase Tropical Surfaces can appear as a limit of a 1-parameter family of smooth complex algebraic surfaces. A phase tropical surface admits a stratified fibration over a smooth tropical surface. We study the real structures compatible with this fibration and give a description in terms of tropical cohomology. As an application we deduce combinatorial criteria for the type of a real structure on a phase tropical surface.