Lowest non-zero vanishing cohomology of holomorphic functions

TL;DR

This paper investigates the lowest non-zero vanishing cohomology of holomorphic functions on complex spaces, providing a geometric method to compute it via nearby curves and applying Lefschetz theorems, with implications for hyperplane arrangements.

Contribution

It introduces a new geometric approach to compute the lowest non-zero vanishing cohomology using nearby curves and Lefschetz theorems, extending results to hyperplane arrangements.

Findings

The lowest non-zero vanishing cohomology can be computed by restriction to a nearby curve.

A Lefschetz type theorem for local fundamental groups is established.

Non-unipotent monodromy in Milnor cohomology vanishes for many hyperplane arrangements.

Abstract

We study the vanishing cycle complex $\varphi_fA_X$ 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 $A_X[d_X]$, we show that the lowest possibly-non-zero vanishing cohomology at $0\in X$ can be calculated by the restriction of $\varphi_fA_X$ 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.