Compact Complex Threefolds Fibred by Two-Dimensional Complex Tori
Abstract
We construct two families of compact complex threefolds fibred over ℙ1 by two-dimensional complex tori. For each positive integer r and each parameter c in a fixed half-plane, the families Yr,c and Zr,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 ℤ/2 and 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 𝒞, and has connected automorphism group ℂ*. For fixed type and index, two members are biholomorphic exactly when their parameters differ by an element of (6/r)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.
Related
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BibTeX
@misc{yang2026compact,
title = {{Compact Complex Threefolds Fibred by Two-Dimensional Complex Tori}},
author = {Xiufan Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29666},
eprint = {2609.29666},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29666},
}
RIS
TY - UNPB
TI - Compact Complex Threefolds Fibred by Two-Dimensional Complex Tori
AU - Xiufan Yang
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.29666
UR - https://arxiv.org/abs/2609.29666
AB - We construct two families of compact complex threefolds fibred over 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 ℤ/2 and 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 𝒞, and has connected automorphism group ℂ^*. For fixed type and index, two members are biholomorphic exactly when their parameters differ by an element of (6/r)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.
ER -