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La conjecture standard de type Hodge prédit la positivité de certaines formes d’intersections sur les cycles algébriques. Cette conjecture est connue en caractéristique 0 via les relations de Hodge-Riemann bilinéaires. Dans cet exposé, on s’intéressera à des nouveaux cas de la conjecture en caractéristique positive, en particulier au cas des puissances de variétés abéliennes de dimension 3.

It is known since Wierzba–Wisniewski’s work in 2003 that a birational map between holomorphic symplectic fourfolds is a composition of Mukai flops. Hu and Yau conjectured that in any dimension, a birational map is a composition of Mukai elementary transformations in codimension two, that is, maps which locally look like a product of a 4-dimensional Mukai flop and the identity on a polydisc, « up to codimension three or higher ». Call  a birational map f from X to X’ a « Hu-Yau transformation » , if there are proper closed subsets Z, Z’ of codimension three or higher such that f induces a Mukai transformation in codimension two between X\Z and X’\Z’. We show that any birational map between irreducible holomorphic symplectic manifolds is a product of Hu-Yau transformations, and give an example showing that a stronger version of the conjecture cannot be true: it is in general not possible to decompose a given map into a product of Mukai elementary transformation after discarding some codimension three closed subsets from the source and the target.