# Genus one mirror symmetry and the arithmetic Riemann--Roch theorem

# Genus one mirror symmetry and the arithmetic Riemann--Roch theorem

**Please note change of location**

Mirror symmetry, in a crude formulation, is usually presented as a correspondence between the Gromov--Witten theory of a Calabi--Yau variety X, and some invariants extracted from the degeneration of Hodge structures of a mirror family of Calabi--Yau varieties. After the physicists Bershadsky--Cecotti--Ooguri--Vafa (henceforth BCOV), this is organised according to the genus of the curves in X we wish to enumerate, and gives rise to an infinite recurrence of differential equations. In this talk, I will present a rigorous mathematical formulation of the BCOV conjecture at genus one, in terms of a lifting of the Grothendieck--Riemann--Roch. I will explain a proof of the conjecture for Calabi--Yau hypersurfaces in projective space, based on the Riemann--Roch theorem in Arakelov geometry. Our results generalise from dimension 3 to arbitrary dimensions previous work of Fang--Lu--Yoshikawa.

This is joint work with G. Freixas and C. Mourougane.