Modular differential equations of the elliptic genus of Calabi–Yau fourfolds

Abstract

We study modular differential equations (MDEs) of the elliptic genus of four-dimensional complex varieties with trivial first Chern class. We construct modular differential equations of orders 3, 4, 5 and 6 with respect to the heat operator for every weak Jacobi form of weight 0 and index 2. We prove that the elliptic genus of a Calabi–Yau fourfold satisfies a MDE of the minimal possible order 3 if and only if its Euler number is equal to 48 or −18. We construct MDEs of order 5 of the elliptic genus of hyperkähler fourfolds of types Hilb2(K3) and Kum2(A).