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Darboux-Halphen-Ramanujan vector field on a moduli of Calabi-Yau manifolds. (English) Zbl 1349.53097

Summary: We obtain an ordinary differential equation \(\mathsf{H}\) from a Picard-Fuchs equation associated with a nowhere vanishing holomorphic \(n\)-form. We work on a moduli space \(\mathsf{T }\) 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 \(\mathsf{H}\) on \(\mathsf{T }\) such that its composition with the Gauss-Manin connection satisfies certain properties. The ordinary differential equation given by \(\mathsf{H}\) is a generalization of differential equations introduced by Darboux, Halphen and Ramanujan.

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
14J15 Moduli, classification: analytic theory; relations with modular forms
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
32G20 Period matrices, variation of Hodge structure; degenerations
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14D07 Variation of Hodge structures (algebro-geometric aspects)
34M45 Ordinary differential equations on complex manifolds
37F75 Dynamical aspects of holomorphic foliations and vector fields
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