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.