Elliptic genus and modular differential equations

We study modular differential equations for the basic weak Jacobi forms in one abelian variable with applications to elliptic genus of Calabi--Yau varieties. show that any $CY_3$ satisfies a equation degree respect heat operator. For $K3$ surface or $CY_5$ is $3$. prove general $CY_4$ its $5$. give examples two operator similar Kaneko--Zagier variable. find type $2$ $3$ second, third and fourth powers theta-series.