The Kuranishi space of complex parallelisable nilmanifolds

Abstract

Abstract. We show that the deformation space of complex parallelisable nilmanifolds can be described by polynomial equations but is almost never smooth. This is remarkable since these manifolds have trivial canonical bundle and are holomorphic symplectic in even dimension. We describe the Kuranishi space in detail in several examples and also analyse when small deformations remain complex parallelisable.

$$\partial \overline{\partial }$$ ∂ ∂ ¯ -Complex symplectic and Calabi–Yau manifolds: Albanese map, deformations and period maps

“…Remark 3.4 -Without assumptions like the degeneration of the Frölicher spectral sequence on the first page, the result is definitely far from true. As shown in [Rol11], most complex parallelisable nilmanifolds have obstructed deformations, for example if they contain an abelian factor. Corollary 3.5 -Let X be a compact complex manifold with trivial canonical bundle whose Frölicher spectral sequence degenerates at the first page.…”

“…In addition, all sufficiently small deformations are again of this type (Section 3). Note that the triviality of the canonical bundle in itself is not strong enough: both properties are known to fail without the ∂∂-Lemma (see [Uen80,Rol11] for examples).…”

Infinitesimal deformations of rational surface automorphisms

“…-The space B X is in general neither reduced nor irreducible (for pathologies, see [46]). If B X is smooth, then we say that X is unobstructed.…”

Generalized deformations and holomorphic Poisson cohomology of solvmanifolds

“…Hence Kuranishi spaces of nilmanifolds with left-invariant complex structures may be singular. In [25], Rollenske studies the singularity of the Kuranishi spaces of complex parallelizable nilmanifolds.…”