Cohomological Milnor Fibrations of Landau--Ginzburg Models

Abstract

We study the global topology of \emph{fundamentally tame} Landau--Ginzburg (LG) models on complete Kähler manifolds. We first show that the superpotential $f$ is surjective onto $\mathbb{C}$, then prove that $f$ restricts to a locally trivial fibration over the set of its regular values, which yields a global generalization of the classical Milnor fibration and provides an Ehresmann-type fibration in the non-proper setting. Assuming that $f$ has finitely many critical points, we use a Morse-theoretic argument to show that the relative homology of $\bigl(M,f^{-1}(c)\bigr)$, for a regular value $c$, is concentrated in the middle degree and has rank equal to the global Milnor number $μ_f$. We further identify the relative homotopy type of the fibers with that of the sublevel sets of $f$ at infinity.Finally, for deformations satisfying a uniform tameness condition on a Stein manifold, we construct the relative cohomology bundles at positive and negative infinity, each equipped with a natural flat Gauss--Manin connection.