On splittings of deformations of pairs of complex structures and holomorphic vector bundles
Hisashi Kasuya, Valto Purho
TL;DR
The paper investigates when the Kuranishi space of a pair $(M,E)$ splits as a product of the Kuranishi spaces of $M$ and $E$. It establishes a product decomposition $Kur_{(M,E)} \cong Kur_M \times Kur_E$ for compact Kähler $M$ with flat Hermitian $E$, and extends this to non-Kähler cases such as complex parallelizable nilmanifolds with trivial $E$, using a DGLA framework. However, it also constructs explicit non-splitting examples on nilmanifolds with abelian complex structures, where interactions between deformations of $M$ and $E$ obstruct the product decomposition (e.g., via concrete Kuranishi series and relations like $t_4 s_3=0$). These results clarify when deformation spaces decouple and when cross-terms persist, enriching the understanding of deformations in complex and nilmanifold geometries.
Abstract
We can show that the Kuranishi space of a pair $(M,E)$ of a compact Kähler manifold $M$ and its flat Hermitian vector bundle $E$ is isomorphic to the direct product of the Kuranishi space of $M$ and the Kuranishi space of $E$. We study non-Kähler case. We show that the Kuranishi space of a pair $(M,E)$ of a complex parallelizable nilmanifold $M$ and its trivial holomorphic vector bundle $E$ is isomorphic to the direct product of the Kuranishi space of $M$ and the Kuranishi space of $E$. We give examples of pairs $(M,E)$ of nilmanifolds $M$ with left-invariant abelian complex structures and their trivial holomorphic line bundles $E$ such that the Kuranishi spaces of pairs $(M,E)$ are not isomorphic to direct products of the Kuranishi spaces of $M$ and the Kuranishi spaces of $E$.