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We consider morphisms $\pi: X \to \mathbb{P}^1$ of smooth projectivevarieties over $\mathbb{C}$. We show that if $\pi$ has at most one singularfibre, then $X$ is uniruled and $\pi$ admits sections. We reach the sameconclusions, but with genus zero multisections instead of sections, if $\pi$has at most two singular fibres, and the first Chern class of $X$ is supportedin a single fibre of $\pi$.  To achieve these result, we use action completed symplectic cohomology groupsassociated to compact subsets of convex symplectic domains. These groups aredefined using Pardon's virtual fundamental chains package for Hamiltonian Floercohomology. In the above setting, we show that the vanishing of these groupsimplies the existence of unirulings and (multi)sections.

Sections and unirulings of families over $\mathbb{P}^1$