Darboux-Halphen-Ramanujan Vector Field on a Moduli of Calabi-Yau Manifolds

الأصل · EN

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