The Kuranishi space of joint deformations of Calabi--Yau manifolds and holomorphic vector bundles

In this paper, we study joint deformations of pairs $(X,E)$, where $X$ is a compact complex manifold and $E\rightarrow X$ is a holomorphic vector bundle. We prove that $(X,E)$ has unobstructed deformations if $X$ is a Fujiki manifold with torsion canonical bundle and $H^2(X,\textrm{End}^0E)=0$, wher