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

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