Lagrangian matching invariants for fibred four-manifolds: I

In a pair of papers, we construct invariants for smooth four-manifolds equipped with ‘broken fibrations’—the singular Lefschetz fibrations of Auroux, Donaldson and Katzarkov—generalising the Donaldson‐Smith invariants for Lefschetz fibrations. The ‘Lagrangian matching invariants’ are designed to be comparable with the Seiberg‐ Witten invariants of the underlying four-manifold; formal properties and first computations support the conjecture that equality holds. They fit into a field theory which assigns Floer homology groups to three-manifolds fibred over S 1 . The invariants are derived from moduli spaces of pseudo-holomorphic sections of relative Hilbert schemes of points on the fibres, subject to Lagrangian boundary conditions. Part I—the present paper—is devoted to the symplectic geometry of these Lagrangians.

在两篇论文中，我们为配备“破碎纤维化”（奥鲁克斯、唐纳森和卡察尔科夫的奇异莱夫谢茨纤维化）的光滑四维流形构造不变量，推广了莱夫谢茨纤维化的唐纳森 - 史密斯不变量。“拉格朗日匹配不变量”旨在与基础四维流形的塞伯格 - 威滕不变量具有可比性；形式性质和首次计算支持等式成立的猜想。它们符合一种场论，该场论将弗洛尔同调群分配给在\(S^1\)上纤维化的三维流形。这些不变量源自纤维上点的相对希尔伯特概型的伪全纯截面的模空间，并受拉格朗日边界条件约束。第一部分——即本文——致力于这些拉格朗日量的辛几何。