A two-parameter family of complex structures on $S^6$

We construct a two-parameter family of complex structures on the standard six-sphere, starting from a rational elliptic surface with singular fibers $II{I}^{\ast },I_1,I_1,I_1$. The resulting torus fibration over ${\mathbb{P}}^1$ has one multiple fiber of multiplicity four, one hexagonal fiber, and two rank-one degenerations. These complex structures are not biholomorphic to the original Alpöge–Claude examples.