Unirationality is the same thing as Rational Connectedness in Characteristic Zero

In this paper we prove that unirationality, rational connectedness and rational chain connectedness coincide for smooth projective varieties over a field $k$ of characteristic zero. Our approach uses the MRC fibration to show that if $X$ is a smooth projective variety, then there exists a variety $\operatorname{MU}(X)$, together with rational maps $\pi: X \dashrightarrow \operatorname{MU}(X)$ and $\lambda: \operatorname{MU}(X) \dashrightarrow \operatorname{MRC}(X)$, such that i) if $\nu: X \dashrightarrow \operatorname{MRC}(X)$, then $\lambda \circ \pi = \nu$ on the appropriate domains; ii) the very general fibres of $\pi$ are unirational; iii) the very general fibres of $\lambda$ are rationally connected but not unirational. We then apply an induction argument to show that $\operatorname{MU}(X)$ is birationally equivalent to $\operatorname{MRC}(X)$.