Let $K$ be a CM field of degree $2n$ over $\mathbf{Q}$ and let $\mathcal{O}_K$ be its ring of integers. Let $\Phi=(\phi_1,\ldots,\phi_n)$ be a CM type of K. Then it is known that complex torus $A:=\mathbf{C}^n/\Phi(\mathcal{O}_K)$ is algebraic. For instance one may construct a Riemann form on $A$ in the following way: choose $\xi\in K$ such that $\xi^2\in K^+$ is totally negative and $\Im(\phi_i(\xi))>0$ for all $i$. Then for a suitable integer $m$ one has that the alternating form $$ E(z,w):=m\sum_i \phi_i(\xi)(\overline{z}_iw_i-z_i\overline{w}_i) $$ is a Riemann form of $A$.

**Q**: Having an abelian varietey $A$ as above is it reasonable to expect it to be the isogeneous to the Jacobian of a curve?

Of course if $n=1$ then the answer is always true but what about $n>1$? Are there some obvious obstructions that prevent $A$ to have a Jacobian origin?