2019/12/30 by Younes Nikdelan, Nikdelan, Younes
Mathematics · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Advanced Mathematical Identities
paper · pdf · doi:10.48550/arxiv.1912.12809
For any positive integer n, we introduce a quasi-homogeneous vector field \textsfD of degree 2 on a moduli space \textsfT of enhanced Calabi-Yau n-folds arising from the Dwork family. By Calabi-Yau quasi-modular forms for Dwork family we mean the elements of the graded ℂ-algebra \widetildeM generated by the components of a particular solution of \textsfD, which are provided with natural weight. Using \textsfD we introduce the derivation D and the Ramanujan-Serre type derivation ∂ on \widetildeM. We show that they are degree 2 differential operators and there exists a proper subspace M⊂ \widetildeM, called the space of Calabi-Yau modular forms, which is closed under ∂. Using the derivation D, we define the Rankin-Cohen brackets for Calabi-Yau quasi-modular forms and prove that the subspace generated by the positive weight elements of M is closed under the Rankin-Cohen brackets.