The Hodge filtration and parametrically prime divisors

TL;DR

This paper investigates the Hodge filtration on meromorphic functions along divisors, providing algebraic formulas and bounds, especially for Euler homogeneous and parametrically prime functions, with numerous examples.

Contribution

It introduces explicit algebraic formulas for the Hodge filtration on sheaves of meromorphic functions, extending to all pieces under specific conditions, and applies these results to various divisor classes.

Findings

Derived a simple algebraic formula for the zeroeth piece of the Hodge filtration

Bounded the first step of the Hodge filtration containing a given function

Computed the Hodge filtration in multiple examples and identified classes of divisors

Abstract

We study the canonical Hodge filtration on the sheaf $\mathscr{O}_X(*D)$ of meromorphic functions along a divisor. For a germ of an analytic function $f$ whose Bernstein-Sato's polynomial's roots are contained in $(-2,0)$, we: give a simple algebraic formula for the zeroeth piece of the Hodge filtration; bound the first step of the Hodge filtration containing $f^{-1}$. If we additionally require $f$ to be Euler homogeneous and parametrically prime, then we extend our algebraic formula to compute every piece of the canonical Hodge filtration, proving in turn that the Hodge filtration is contained in the induced order filtration. Finally, we compute the Hodge filtration in many examples and identify several large classes of divisors realizing our theorems.