  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.
