Preprint A755/2014
Darboux-Halphen-Ramanujan Vector Field on a Moduli of Calabi-Yau Manifolds
Younes Nikdelan
Keywords: Darboux-Halphen-Ramanujan vector field | Hodge structure | Picard-Fuchs equation | Gauss-Manin connection

In this paper we obtain an ordinary differential
equation ${\sf H}$ from a  Picard-Fuchs equation  associated with a nowhere vanishing
holomorphic $n$-form. We work on a moduli space ${\sf T }$ constructed from  a Calabi-Yau manifold $W$ together
with a basis of the middle complex de Rham cohomology of $W$. We verify the existence of
a unique vector field ${\sf H}$ on ${\sf T }$  such that its composition with the Gauss-Manin connection satisfies certain
properties. The ordinary differential equation given by
${\sf H}$ is a generalization of differential equations introduced by Darboux, Halphen and Ramanujan.