The question is kind of self contained, but let me develop a bit further.

Assume K is a CM field of degree $2g$, that is, a quadratic imaginary extension of a totally real field. A CM type of $K$ is a set $\Phi$ consisting of $g$ complex embeddings of $K$ such that $\mathrm{Hom}(K, \mathbb{C})=\Phi \cup \overline{\Phi}$.

Given $(K, \Phi)$, there always exists a $g$-dimensional abelian variety $A$ such that $\mathrm{End}(A) \otimes \mathbb{Q}=K$ and that $K$ acts on $H^0(A, \Omega^1_A)$ through $\Phi$. One easily constructs as a complex torus, starting from the embedding $K \subset \mathbb{C}^g$ given by $\Phi$.

The following question seems much more subtle:

*Is there always such an abelian variety of the form $A=\mathrm{Jac}(C)$ for a smooth projective curve $C$?*