Preprint C178/2014
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

This research was intended as an attempt to obtain an ordinary
differential equation $\rm H$ from a linear differential equation
$L$. We work on a Calabi-Yau $n$-fold $W$ whose complex deformation is
one dimensional and middle complex de Rham cohomology
$H^n_{\rm dR}(W;\mathbb{C})$ is $(n+1)$-dimensional. Moreover, we suppose
that the Picard-Fuchs equation $L$ associated with the unique
nowhere vanishing holomorphic $n$-form $\omega$ on $W$ is of
order $n+1$. As a first result, we prove that $L$ is self-dual.

Next, we define ${\rm \textsf{T} }$ to be the moduli of $W$ together with a
basis of $H^n_{\rm dR}(W;\mathbb{C})$, where the basis is required to be
compatible with the Hodge filtration of $H^n_{\rm dR}(W;\mathbb{C})$;
furthermore, the intersection form matrix in this basis is
constant. Then in our second result, we verify the existence of
a unique vector field $\rm H$ on ${\rm \textsf{T} }$ that satisfies certain
properties. Indeed, the ordinary differential equation given by
$\rm H$ is a generalization of differential
equations introduced by Darboux, Halphen and Ramanujan.