In recent joint work (w. Gerard Freixas and Christophe Mourougane) we proved a mirror symmetry-statement for genus one Gromov-Witten invariants of Calabi-Yau hypersurfaces in projective space. The original conjecture was formulated by string theorists. A central tool was a reformulation of the conjecture, using an metric version of the Riemann-Roch.
In this talk I will focus on a formulation of a more mathematical version of the conjecture, together with a list of examples where it is verified. There are two main ingredients, namely the limit Hodge structure of a maximally degenerate family, and a lifting of the (codimension 1)-version of the Grothendieck-Riemann-Roch theorem to the level of line bundles. The latter ingredient is ongoing joint work Gerard Freixas.