2020/08/24 by Morihiko Saito, Saito, Morihiko · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2008.10529
openalex publication_date 2020/08/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the vanishing cycle complex φfAX for a holomorphic function f on a reduced complex analytic space X with A a Dedekind domain (for instance, a localization of the ring of integers of a cyclotomic field, where the monodromy eigenvalue decomposition may hold after a localization of A). Assuming the perversity of the shifted constant sheaf AX[dX], we show that the lowest possibly-non-zero vanishing cohomology at 0∈ X can be calculated by the restriction of φfAX to an appropriate nearby curve in the singular locus Y of f, which is given by intersecting Y with the intersection of sufficiently general hyperplanes in the ambient space passing sufficiently near 0. The proof uses a Lefschetz type theorem for local fundamental groups. In the homogeneous polynomial case, a similar assertion follows from Artin's vanishing theorem. By a related argument we can show the vanishing of the non-unipotent monodromy part of the first Milnor cohomology for many central hyperplane arrangements with ambient dimension at least 4.