Séminaire de géométrie - Salle de conférences du CMLS - 14h

Titre: Mixed Hodge structures and heights associated to algebraic cycles

Résumé: In abstract Hodge theory, Deligne’s delta splitting measures how far a mixed Hodge structure is from being split as a real mixed Hodge structure. An allied notion, developed by S. Bloch, R.Hain et al., is that of a height for a special class of mixed Hodge structures called Biextensions. The idea of a Biextension is closely related to algebraic cycles homologous to zero. Given two such cycles in complementary codimensions in an ambient smooth and projective variety, a certain cohomology group associated to the pair provides an example of a Biextension-type mixed Hodge structure. The height associated with such a Biextension has been well studied and has been an active area of research for the past few decades. In an ongoing project, the speaker, along with J. I. Burgos Gil and G. Pearlstein, has developed a theory of mixed Hodge structures and heights associated with Bloch’s higher cycles that generalizes the above study of Biextensions (doi.org/10.1112/plms.12443 and arXiv:2410.17167v2 [math.AG]). This talk will have two sections: In the first, I will give a gentle introduction to the theory of archimedean height pairing, ending with a simple example on the projective line. The second section will be a survey of the current state of the art, with focus on the above article and preprint.