Modular differential equations of minimal orders of the elliptic genus of Calabi--Yau varieties

2025/09/30 by Adler, Dmitrii, Gritsenko, Valery
#11F50 #17B69 #32W50 #58J26 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.25954

Abstract

We study modular differential equations (MDEs) of high orders for weak Jacobi forms and find necessary conditions for weak Jacobi forms to satisfy MDEs of order 3 with respect to the heat operator. We investigate all possible MDEs for the elliptic genus of six-dimensional manifolds with a trivial first Chern class. We prove that the minimal possible order of the MDE for the elliptic genus of a strict six-dimensional Calabi--Yau variety is four, and find MDEs of order 7 for hyperkähler varieties of dimension 6. The latter MDEs correspond to the generic case. The non-generic weak Jacobi forms of weight 0 and index 3 form a divisor that contains two cubic plane curves in the coefficient space.

Citations

• Modular linear differential equations of fourth order and minimal W-algebras
• On modular forms arising from a differential equation of hypergeometric type
• Elliptic genus of Calabi-Yau manifolds and Jacobi and Siegel modular forms
• Automorphic Forms and Lorentzian Kac-Moody Algebras. Part II

Related

• Elliptic genus of Calabi-Yau manifolds and Jacobi and Siegel modular forms
• About a class of Calabi-Yau differential equations
• Anomaly cancellation formulas and E₈ bundles for almost complex manifolds
• Elliptic curves in hyper-Kähler varieties
• Elliptic genera, real algebraic varieties and quasi-Jacobi forms