Let $f:Y\rightarrow\mathbb{P}^1$ be an elliptic $K3$ surface, then the holomorphic 2-form $\Omega\_Y$ vanishes when restricted to an elliptic fiber $f^{-1}(b)$ with $b\in\mathbb{P}^1$. After a hyperkahler rotation, we get a new $K3$ surface $X$, and the map $f:X\rightarrow S^2$ becomes a special Lagrangian fibration with respect to the Calabi-Yau structure $(\omega\_X,\Omega\_X)$ on $X$. For $X$, one can then talk about its SYZ mirror $X^\vee$. Note that we have an $S^1$ family of choice of such $X$ so that $f:X\rightarrow S^2$ is a special Lagrangian fibration. By choosing the symplectic structure $\omega\_X$ on $X$ appropriately, the mirror symmetry of $X$ can be interpreted by a hyperkahler rotation. Applying hyperkahler rotation we get the mirror $X^\vee$, which is still a $K3$ surface, the underlying differentiable structure is unchanged but with a different complex structure. We can choose $\omega\_X$ so that $f:X^\vee\rightarrow S^2$ is still a special Lagrangian fibration with respect to $\omega\_{X^\vee}$ and $\Omega\_{X^\vee}$.


Note that in general $X^\vee$ may not be an elliptic $K3$ surface. My question is when will $X^\vee$ be an elliptic $K3$?



When will the mirror of a K3 surface be an elliptic K3?
A K3 surface admits an elliptic fibration if and only if there is an isotropic vector in the Neron-Severi lattice (the elliptic fiber corresponds to the isotropic vector after a sequence of reflections by $(-2)$-curves). So the mirror K3 surface is elliptic if and only if there is an isotropic vector orthogonal to the period vector $\Omega\_{X^\vee}=\omega\_X+i Re\Omega\_X$. 


