On splittings of deformations of pairs of complex structures and holomorphic vector bundles
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 $E$.