on a family of CM Hodge structures I am having a look at the paper "Hodge structures of type $(n, 0, \ldots, 0, n)$" by Totaro. At the very beginning he says that a Hodge structure of type $(1, 0, 1)$ has always complex multiplication. 
Here, type $(1, 0, 1)$ means that $V$ is a pure $\mathbb{Q}$-Hodge structure of weight 2 with Hodge decomposition $V_{\mathbb{C}}=V^{2, 0} \oplus V^{0, 2}$ and $V^{2, 0}$ one dimensional. 
"Complex multiplication" means in this case that there is an imaginary quadratic number field $K$ acting on $V$ by endomorphisms of Hodge structures. 
Why is is true?
My guess is that you can renumber the pieces of the Hodge decomposition to get a weight one Hodge structure of dimension 2, which should correspond to an elliptic curve.
1) Does this show that the Hodge structure $V$ is automatically polarizable?
2) Why does the elliptic curve constructed in that way have complex multiplication?
3) Does there always exists a K3 surface $X$ such that $V$ is the transcendental part of the  singular cohomology $H^2(X(\mathbb{C}, \mathbb{Q})$? 
 A: 1) Such a Hodge structure is always polarizable: if $\omega $ is a nonzero element of $V^{2,0}$, you just have to choose the number $\langle \omega ,\bar{\omega }\rangle=\lambda $. Choose $\omega $ such that $\omega +\bar{\omega }$ belongs to $V$, and take $\lambda \in\mathbb{Q}$.
2) Choose a lattice $V_{\mathbb{Z}}$ in $V$, so indeed $V^{2,0}/V_{\mathbb{Z}}$ is an elliptic curve $E$; write as usual $V_{\mathbb{Z}}=\mathbb{Z}+\mathbb{Z}\tau $, $\tau \in\mathbb{H}$. The polarization gives us a 
hermitian form $H$ on $V^{2,0}$ such that the real part of $H$ takes rational values on $V$. Put $H(1,1)=a\ (\in\mathbb{Q})$; then $q:=\tau \bar{\tau }=\frac{1}{a}H(\tau ,\tau )$ and $p:=\tau +\bar{\tau }=\frac{2}{a}\mathrm{Re}\,H(1,\tau )$ are rational, so $\tau $ satisfies the quadratic equation $\tau ^2-p\tau +q=0$, hence $E$ is CM.
3) I think yes, the Kummer surface obtained by desingularizing $(E\times E)/\{\pm 1\} $ does the job. This is in a paper by Shioda and Inose,
On singular K3 surfaces, Complex analysis and algebraic geometry, pp. 119-136. Iwanami Shoten, Tokyo, 1977. 
