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 $C{Y}_3$ satisfies a differential equation of degree one with respect to the heat operator. For a $K3$ surface or any $C{Y}_5$ the degree of the differential equation is $3$. We prove that for a general $C{Y}_4$ 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.