Twisted logarithmic complexes of positively weighted homogeneous divisors

TL;DR

This paper investigates the conditions under which the twisted logarithmic comparison theorem holds for positively weighted homogeneous divisors, linking it to properties of D-module annihilators and Bernstein-Sato polynomials.

Contribution

It establishes new criteria for the logarithmic comparison theorem in the context of weighted homogeneous divisors using D-module theory and Bernstein-Sato polynomial analysis.

Findings

LCT holds if and only if the annihilator of f^{α-1} is generated by first-order operators.

When the divisor is defined by a homogeneous polynomial with isolated weighted homogeneous singularities, LCT is equivalent to -1 being the unique integral root of the Bernstein-Sato polynomial.

A simple proof of LCT for hyperplane arrangements is provided under certain residue conditions, utilizing Castelnuovo-Mumford regularity.

Abstract

For a rank 1 local system on the complement of a reduced divisor on a complex manifold $X$, its cohomology is calculated by the twisted meromorphic de Rham complex. Assuming the divisor is everywhere positively weighted homogeneous, we study necessary or sufficient conditions for a quasi-isomorphism from its twisted logarithmic subcomplex, called the logarithmic comparison theorem (LCT), by using a stronger version in terms of the associated complex of $D_X$-modules. In case the connection is a pullback by a defining function $f$ of the divisor and the residue is $\alpha$, we prove among others that if LCT holds, the annihilator of $f^{\alpha-1}$ in $D_X$ is generated by first order differential operators and $\alpha-1-j$ is not a root of the Bernstein-Sato polynomial for any positive integer $j$. The converse holds assuming either of the two conditions in case the associated complex of $D_X$-modules is acyclic except for the top degree. In the case where the local system is constant, the divisor is defined by a homogeneous polynomial, and the associated projective hypersurface has only weighted homogeneous isolated singularities, we show that LCT is equivalent to that $-1$ is the unique integral root of the Bernstein-Sato polynomial. We also give a simple proof of LCT in the hyperplane arrangement case under appropriate assumptions on residues, which is an immediate corollary of higher cohomology vanishing associated with Castelnuovo-Mumford regularity. Here the zero-extension case is also treated.