Deformations of Complex Nilmanifolds and Torus Bundles

Definitions and mathematical statement

Let $M=\Gamma\backslash G$ be a compact nilmanifold, where $G$ is a nilpotent real Lie group and $\Gamma$ a discrete cocompact subgroup, equipped with a left-invariant integrable complex structure $J$. Such a structure is induced by an endomorphism of the Lie algebra with $J^2=-1$ and vanishing Nijenhuis tensor. The usual connected/simply-connected convention on $G$ is not explicit in the source and should be confirmed.

The source asks three tasks:

1. Determine the small deformations of the complex manifold $(M,J)$. This concerns its full local complex-deformation space, not merely the subfamily of left-invariant complex structures. In particular, criteria ensuring that all small deformations remain left invariant are only part of the requested picture.
2. Determine when $(M,J)$ has a geometric description as an iterated principal holomorphic torus bundle. In the usual sense this is a finite tower of holomorphic principal bundles with compact complex torus fibers, ending at a point or the equivalent terminal-torus convention. Principal means translation structure group; an arbitrary holomorphic torus fibration is not silently identified with a principal bundle.
3. Determine all deformations in the large of such bundles. This asks for the global deformation class in the intended moduli/deformation sense, not just a neighborhood of one complex structure.

The record does not prescribe a Kuranishi or other local deformation model, marked versus unmarked equivalence, the global deformation relation, the allowed torus-bundle tower, or hypotheses involving the lattice and complex Lie-algebra data. Nor does it assert that every deformation remains invariant, nilmanifold-like, or a principal torus bundle. Those are potential conclusions to analyze, not assumptions to add.

The supplied assessment attributes criteria for small-deformation invariance and substantial large-deformation results to Rollenske, with a fairly complete complex-dimension-three picture. The source's general three-part task is not restricted to dimension three or identified with those reported partial theorems.

Short English statement

The entry asks for local deformations of invariant complex nilmanifolds, criteria for an iterated principal holomorphic torus-bundle description, and the full large-deformation class of such bundles. Full complex deformations must remain distinct from the invariant subfamily.

Sources

• Local UnsolvedMath registry, permanent ID 30000857, code catalogue entry, "Deformations of Complex Nilmanifolds and Torus Bundles." Proposed year 2007; proposer unspecified.
• Komplexe Algebraische Geometrie (2008), Oberwolfach Reports, pp. 2791--2850, DOI 10.4171/owr/2007/47; https://doi.org/10.4171/owr/2007/47. The displayed report year 2008 differs from the DOI and proposed year 2007.
• Supplied "Rollenske, nilmanifolds and deformations": https://doi.org/10.1112/plms/pdp001.
• Supplied "Rollenske, Lie-algebra Dolbeault cohomology": https://arxiv.org/abs/0709.0467.
• Statement status corrected_verified; the supplied note says official-report content was checked and wording/notation normalized without changing the claim.
• Literature triage checked 2026-08-21: partially solved. It reports Rollenske's small-deformation invariance criteria, large-deformation results for iterated principal holomorphic torus bundles, and a fairly complete dimension-three picture, while the full classification is broader. Exact hypotheses and current status require independent verification. Created 2026-08-25; updated 2026-09-16.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; underlying report and literature provenance remain as supplied. Unreviewed statement-only draft.