Universal affine bundles for compact complex manifolds

We construct a universal affine bundle $E$ over a compact complex manifold $X$ equipped with a probability measure $\mu$, which is affine over the vector space $H_{1,1}(X;\mathbb{R})$, and satisfies a number of natural properties. The bundle $E$ carries a natural action of the automorphism group of $(X,\mu)$, and provides potentials for all $dd^c$-closed $(1,1)$-forms on $X$. We relate our construction to lifts of the action of the automorphism group of a Calabi--Yau manifold to its universal torsor.