In mathematics, an abelian variety A defined over a field K is said to have CM-type if it has a large enough commutative subring in its endomorphism ring End(A).
The problem is at a deeper level of abstraction, because it is much harder to manipulate analytic functions of several complex variables.
The formal definition is that the tensor product of End(A) with the rational number field Q, should contain a commutative subring of dimension 2d over Q.
The CM-type is a description of the action of a (maximal) commutative subring L of EndQ(A) on the holomorphic tangent space of A at the identity element.
Spectral theory of a simple kind applies, to show that L acts via a basis of eigenvectors; in other words L has an action that is via diagonal matrices on the holomorphic vector fields on A.