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 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 (M,f−1(c)), for a regular value c, is concentrated in the middle degree and has rank equal to the global Milnor number μf. 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.
Related
- Milnor fibrations on rho-tubes and rho-spheres for real analytic map germs
- Existence of weak Landau-Ginzburg models for weighted complete intersections
- Topological defect trains excited by a boundary discontinuity in the two-dimensional complex Ginzburg-Landau equation
- Landau--Ginzburg mirror symmetry for ADE singularities
- Global minimality of the degree-one Ginzburg-Landau vortex solution in the unit ball in dimensions n ≥ 2
BibTeX
@misc{fan2026cohomological,
title = {{Cohomological Milnor Fibrations of Landau--Ginzburg Models}},
author = {Huijun Fan and Jieran Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25825},
eprint = {2610.25825},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25825},
}
RIS
TY - UNPB
TI - Cohomological Milnor Fibrations of Landau--Ginzburg Models
AU - Huijun Fan
AU - Jieran Wang
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.25825
UR - https://arxiv.org/abs/2610.25825
AB - 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 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 (M,f^−1(c)), 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.
ER -