We consider morphisms \(\pi : X \to \mathbb{P}^{1}\) of smooth projective varieties over \(\mathbb{C}\). We show that if π has at most one singular fibre, then X is uniruled and π admits sections. We reach the same conclusions, but with genus zero multisections instead of sections, if π has at most two singular fibres, and the first Chern class of X is supported in a single fibre of π.

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.

我们考虑\ (\mathbb{C}\)上平滑射影簇的态射\(\pi : X \to \mathbb{P}^{1}\)。我们证明，如果π至多有一根奇异纤维，则X是无规的并且π允许分段。我们得出相同的结论，但使用属零多节而不是节，如果π至多有两个奇异纤维，并且X的第一个陈氏类由π的单个纤维支持。

为了实现这些结果，我们使用与凸辛域的紧子集相关的动作完成辛上同调群。这些群是使用 Pardon 的哈密顿弗洛尔上同调的虚拟基本链包定义的。在上述设置中，我们表明这些群体的消失意味着单规则和（多）部分的存在。