[@Mum].

[**Remark.**]{} Shimura proved that there exist ${\mathbf{z}}\in \mathfrak{h}^g$ for which $\operatorname{End}_\QQ(X_{{\mathbf{z}}}) = K$, see [@Shimura Theorem 5, p. 176] or [@BL Theorem 9.9.1, p. 274]. Using this result, together with the fact that $K$ contains no nontrivial idempotents, and [@BL Theorem 5.3.2 p. 123] we conclude that there exists simple abelian varieties for which Theorem \[existence:prop\] applies.

### Real multiplication and semi-homogeneous vector bundles

Let $K$ be a totally real number field of degree $e$ over $\QQ$ and let $\sigma_1, \dots, \sigma_e$ denote the embeddings of $K$. Let $g$ be a positive integer which is divisible by $e$, and let $m := \frac{g}{e}$. Let $\mathfrak{h}^{\oplus e}_m$ denote the $e$-fold product of the Siegel upper half space $\mathfrak{h}_m$. Finally, let $\rho \colon K \rightarrow M_g(\CC)$ be the representation of $\QQ$-algebras defined by $$\eta \mapsto \operatorname{diag}(\sigma_1(\eta),\dots, \sigma_e(\eta)) \otimes \mathbf{1}_m \text{,}$$ where the tensor product denotes the Kronecker product of matrices.

Before stating a more general version of Theorem \[existence:prop\] we recall that if $X = \CC^g / \Lambda$ is a complex torus then $\operatorname{NS}(X)_\QQ$ can be identified with the collection of Hermitian forms $H \colon \CC^g \times \CC^g \rightarrow \CC$ whose imaginary part is rational valued on $\Lambda$. Using this identification 