Compact Complex Threefolds Fibred by Two-Dimensional Complex Tori

Abstract

We construct two families of compact complex threefolds fibred over $\mathbb P^1$ by two-dimensional complex tori. For each positive integer $r$ and each parameter $c$ in a fixed half-plane, the families $Y_{r,c}$ and $Z_{r,c}$ have two multiple fibres of multiplicities $(4,6)$ and $(3,6)$, respectively, and one reduced normal-crossings fibre whose normalization consists of $r$ del Pezzo surfaces of degree six. Their fundamental groups are $\mathbb Z/2$ and $\mathbb Z/3$, respectively, and both have Euler characteristic $2r$. We compute their integral homology; the index-one members are rational homology six-spheres. Every member has algebraic dimension one, lies outside Fujiki's class $\mathcal C$, and has connected automorphism group $\mathbb C^*$. For fixed type and index, two members are biholomorphic exactly when their parameters differ by an element of $(6/r)\mathbb Z$. Each underlying smooth six-manifold therefore supports uncountably many pairwise non-biholomorphic complex structures. The arithmetic input classifies finite monodromy pairs and their integral lifts to a parabolic Jacobi group acting on a rank-four lattice. A quadratic form controls integral lifting and, on the rank-two square-zero locus, computes the monodromy saturation index. We prove that the relevant invariant tensors persist under finite-index restriction, which extends the construction to arbitrary compact base curves. We also construct translation twists after base change, establish a local periodic toroidal quotient theorem in arbitrary dimension, and determine the additive Bockstein spectral sequences of the original threefolds.