On the topology of fibers of complex polynomial maps

2026/08/05 by Laurentiu Maxim, John Messina
Mathematics · #math.AG #math.AT

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arxiv created 2026/08/05 · arxiv updated 2026/08/06

Abstract

We survey known results on the cohomology of fibers of complex polynomial maps, with particular emphasis on vanishing ranges, and establish new results on the vanishing cohomology of a polynomial at a bifurcation value. We also derive upper bounds for the first possibly nonvanishing Betti number of general and atypical fibers in terms of local singularity invariants. These results extend several theorems of Tibăr, Siersma, Dimca, and others from the case of isolated singularities, including singularities at infinity, to polynomial maps with arbitrary singularities.

Citations

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• Betti Bounds of Polynomials
• Sheaves in Topology
• Singularity exchange at the frontier of the space
• Singularities and Topology of Meromorphic Functions
• On the connectivity of complex affine hypersurfaces II
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