Quasi modular forms for variation of Calabi-Yau varieties

Abstract: Ramanujan relations between Eisenstein series can be seen as ordinary differential equations on the moduli space of elliptic curves equipped with elements in their first de Rham cohomologies which are not of the first kind. This gives us a geometric interpretation of quasi modular forms associated to elliptic curves. Another example of such differential equations is the Halphen equation which can can be derived from Gauss hypergeometric function. In this talk I present a new ordinary differential equation in seven dimensions related to the variation of certain Calabi-Yau varieties. These varieties appear in mathematical physics and are mirror dual to generic quintic Calabi-Yau threefolds. We construct the theory of quasi modular functions associated to these varieties in such a way that it includes the generating series for counting virtual number of rational curves on generic quintic hypersufaces introduced by physicists Candelas, de la Ossa, Green and Parkes in 1991. In this way we also reformulate and realize a problem of P. Griffiths around 1978 for the the moduli of polarized Hodge structures.