Elliptic genus and modular differential equations

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 $CY_3$ satisfies a differential equation of degree one with respect to the heat operator. For a $K3$ surface or any $CY_5$ the degree of the differential equation is $3$. We prove that for a general $CY_4$ 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 operator 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.