On splittings of deformations of pairs of complex structures and holomorphic vector bundles

TL;DR

This paper investigates the structure of Kuranishi spaces for pairs of complex manifolds and holomorphic vector bundles, showing conditions under which these spaces decompose into products and providing counterexamples.

Contribution

It extends the understanding of Kuranishi space decompositions from Kähler to non-Kähler cases, especially for nilmanifolds and specific complex structures.

Findings

Kuranishi space of a pair (M,E) can be isomorphic to a product in Kähler cases.

For certain nilmanifolds, the product decomposition also holds.

Counterexamples show that the product structure does not always apply in non-Kähler cases.

Abstract

We can show that the Kuranishi space of a pair $(M,E)$ of a compact K\"ahler 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-K\"ahler 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 $E$.