arXiv:2209.00038v1 [math.AG] 31 Aug 2022 ELLIPTIC GENUS AND MODULAR DIFFERENTIAL EQUATIONS DMITRII ADLER AND VALERY GRITSENKO Abstract. We study modular differential equations for the basic weak Jacobi forms in one abelian variable with applications to the elliptic genus of Calabi–Yau varieties. We show that the elliptic genus of any CY3 satisfies a differential equation of degree one with respect to the heat operator. For a K3 surface or any CY5 the degree of the differ- ential equation is 3. We prove that for a general CY4 its elliptic genus satisfies a modular differential equation of degree 5. We give examples of differential equations of degree two with respect to the heat oper- ator similar to the Kaneko–Zagier equation for modular forms in one variable. We find modular differential equations of Kaneko–Zagier type of degree 2 or 3 for the second, third and fourth powers of the Jacobi theta-series. 1. Elliptic genus and Jacobi modular forms 1.1. Elliptic genus of complex varieties with c 1 =0. Jacobi modular forms appear in geometry and physics as special partition functions. For example, the elliptic genus of any complex compact variety of dimension d with trivial first Chern class is a weak Jacobi form of weight 0 and index d/2 with integral Fourier coefficients (see [18] in the context of N =2 superconformal field theory, [8] in the context of Jacobi forms, [22] in the context of elliptic homology). Differential equations for modular forms in one variable τ ∈ H were con- sidered in the first half of the 20th century by Ramanujan and Rankin (see [23]). In this paper we find differential equations for the elliptic genera of Calabi–Yau varieties of small dimensions. For a Jacobi form of index d/2 one has to consider the heat operator H (d) =4πid ∂ ∂τ − ∂ 2 ∂z 2 instead of the differentiation d dτ where τ ∈ H and z ∈ C. We prove that the elliptic genus of any CY 3 satisfies a differential equation of degree one with respect to the heat operator. We find a modular differential equation of degree 3 in H (1) for a K3 surface and in H ( 5 2 ) for any CY 5 . We prove that for a general CY 4 there exists a modular differential equation of degree 5 with respect to the heat operator. We note that modular differential equations are important in the descrip- tion of the characters of a conformal field theory. For N = 2 superconformal Date : September 2, 2022. 2010 Mathematics Subject Classification. 11F50, 17B69, 32W50, 58J26. 1