On genus one mirror symmetry in higher dimensions and the BCOV conjectures
Résumé
The mathematical physicists Bershadsky--Cecotti--Ooguri--Vafa (BCOV) proposed, in a seminal article from '94, a conjecture extending genus zero mirror symmetry to higher genera. With a view towards a refined formulation of the Grothendieck--Riemann--Roch theorem, we offer a mathematical description of the BCOV conjecture at genus one. As an application of the arithmetic Riemann--Roch theorem of Gillet--Soul\'e and of our previous results on the BCOV invariant, we establish this conjecture for Calabi--Yau hypersurfaces in projective spaces. Our contribution takes place on the $B$-side, and together with the work of Zinger on the $A$-side, it provides the first complete examples of the mirror symmetry program in higher dimensions. The case of quintic threefolds was studied by Fang--Lu--Yoshikawa.
Our approach also lends itself to arithmetic considerations of the BCOV invariant, and we study a Chowla--Selberg type theorem expressing it in terms of special $\Gamma$ values for certain Calabi--Yau manifolds with complex multiplication.
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