Twisted logarithmic complexes of positively weighted homogeneous divisors

For a rank 1 local system on the complement of a 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 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. If the connection is a pullback by a defining function $f$ of the divisor with residue $\alpha$, we show that LCT implies that 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 in case the associated complex of $D_X$-modules is acyclic except for the top degree. We also give a simple proof of LCT in the hyperplane arrangement case under appropriate assumptions on residues as an immediate corollary of higher cohomology vanishing associated with Castelnuovo-Mumford regularity, where the zero-extension case is also treated