Canonical complex extensions of K\"ahler manifolds

Given a complex manifold $X$, any K\"ahler class defines an affine bundle over $X$, and any K\"ahler form in the given class defines a totally real embedding of $X$ into this affine bundle. We formulate conditions under which the affine bundles arising this way are Stein and relate this question to other natural positivity conditions on the tangent bundle of $X$. For compact K\"ahler manifolds of non-negative holomorphic bisectional curvature, we establish a close relation of this construction to adapted complex structures in the sense of Lempert--Sz\H{o}ke and to the existence question for good complexifications in the sense of Totaro. Moreover, we study projective manifolds for which the induced affine bundle is not just Stein but affine and prove that these must have big tangent bundle. In the course of our investigation, we also obtain a simpler proof of a result of Yang on manifolds having non-negative holomorphic bisectional curvature and big tangent bundle.