Séminaires : Séminaire de géométrie algébrique

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Le jeudi à 14h.
septembre-décembre Sophie-Germain, janvier-mars ENS, avril-juin Jussieu

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Résumé : Let X be a complex projective surface. A real form of X is a real projective variety W, whose Cartesian product with SpecC over SpecR recovers X.Two real forms are considered isomorphic if they are isomorphic over SpecR. A natural question is to ask how many non-isomorphic real forms can be attributed to a fixed complex projective variety X: In particular, are there finitely many ? As soon as X admits at least one real form, this question boils down to counting non-conjugate involutions in a group naturally associated to X. In this talk, we emphasize two aspects of this counting problem: We first explain why varieties satisfying the Kawamata-Morrison cone conjecture (such as K3 surfaces, Enriques surfaces, abelian surfaces) have finitely many real forms; we then describe a smooth blow-up of an Enriques surface at one point,which is endowed with infinitely many real forms.

This is joint work with Tien-Cuong Dinh, Hsueh-Yung Lin, Keiji Oguiso, Long Wang, and Xun Yu.