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On strong Euler-homogeneity and Saito-holonomicity for complex hypersurfaces. Applications to a conjecture on free divisors

Abraham del Valle Rodríguez

TL;DR

The article develops a framework linking strong Euler-homogeneity, Saito-holonomicity, and the Logarithmic Comparison Theorem (LCT) through Fitting ideals of Jacobian- and logarithmic-derivation modules. It extends these criteria to formal power series and uses them to establish new cases where LCT implies strong Euler-homogeneity for free divisors, including when SEH holds off a discrete set, for weak Koszul-free divisors, and in dimension four, aided by a formal intrinsic structure theorem. It also analyzes linear free divisors, providing a higher-dimensional counterexample to LCT, symmetry of the $b$-function, and a targeted result showing LCT implies SEH for linear divisors in dimension five, together with dimensional-induction implications. Overall, the work strengthens the connections between LCT, Euler-homogeneity, and Saito-holonomicity via a rigorous algebraic framework and a formal–analytic bridge, advancing understanding of conjectures on free divisors.

Abstract

We first develop some criteria for a general divisor to be strongly Euler-homogeneous in terms of the Fitting ideals of certain modules. We also study new variants of Saito-holonomicity, generalizing Koszul-free type properties and characterizing them in terms of the same Fitting ideals. Thanks to these advances, we are able to make progress in the understanding of a conjecture from 2002: a free divisor satisfying the Logarithmic Comparison Theorem (LCT) must be strongly Euler-homogeneous. Previously, it was known to be true only for ambient dimension $n \leq 3$ or assuming Koszul-freeness. We prove it in the following new cases: assuming strong Euler-homogeneity outside a discrete set of points; assuming the divisor is weakly Koszul-free; for $n=4$; for linear free divisors in $n=5$. Finally, we refute a conjecture stating that all linear free divisors satisfy LCT, are strongly Euler-homogeneous and have $b$-functions with symmetric roots about $-1$.

On strong Euler-homogeneity and Saito-holonomicity for complex hypersurfaces. Applications to a conjecture on free divisors

TL;DR

The article develops a framework linking strong Euler-homogeneity, Saito-holonomicity, and the Logarithmic Comparison Theorem (LCT) through Fitting ideals of Jacobian- and logarithmic-derivation modules. It extends these criteria to formal power series and uses them to establish new cases where LCT implies strong Euler-homogeneity for free divisors, including when SEH holds off a discrete set, for weak Koszul-free divisors, and in dimension four, aided by a formal intrinsic structure theorem. It also analyzes linear free divisors, providing a higher-dimensional counterexample to LCT, symmetry of the -function, and a targeted result showing LCT implies SEH for linear divisors in dimension five, together with dimensional-induction implications. Overall, the work strengthens the connections between LCT, Euler-homogeneity, and Saito-holonomicity via a rigorous algebraic framework and a formal–analytic bridge, advancing understanding of conjectures on free divisors.

Abstract

We first develop some criteria for a general divisor to be strongly Euler-homogeneous in terms of the Fitting ideals of certain modules. We also study new variants of Saito-holonomicity, generalizing Koszul-free type properties and characterizing them in terms of the same Fitting ideals. Thanks to these advances, we are able to make progress in the understanding of a conjecture from 2002: a free divisor satisfying the Logarithmic Comparison Theorem (LCT) must be strongly Euler-homogeneous. Previously, it was known to be true only for ambient dimension or assuming Koszul-freeness. We prove it in the following new cases: assuming strong Euler-homogeneity outside a discrete set of points; assuming the divisor is weakly Koszul-free; for ; for linear free divisors in . Finally, we refute a conjecture stating that all linear free divisors satisfy LCT, are strongly Euler-homogeneous and have -functions with symmetric roots about .
Paper Structure (13 sections, 34 theorems, 36 equations)

This paper contains 13 sections, 34 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. About products with smooth factors
  4. From convergent to formal definitions
  5. About singular derivations
  6. Strong Euler-homogeneity around a point
  7. Saito-holonomic type properties
  8. Extending the criteria to formal power series
  9. Proving new cases of Conjecture \ref{['ConjLCT-SEH']}
  10. Assuming strong Euler-homogeneity outside a discrete set of points
  11. Assuming weak Koszul-freeness
  12. Assuming ambient dimension $n=4$
  13. The case of linear free divisors

Key Result

Lemma 2.2

Let $f$ be a reduced local equation of a divisor $D \subset X$ at a point $p \in D$. Then, $f$ is a product at $p$ if and only if it admits a non-singular logarithmic derivation, that is, if $\mathop{\mathrm{Der}}\nolimits_{X,p}(-\log D)$ is not contained in $\mathfrak{m}_p \mathop{\mathrm{Der}}\nol

Theorems & Definitions (95)

  • Definition 1.1
  • Conjecture 1.2
  • Example 1.3
  • Definition 1.4
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Remark 2.4
  • Definition 2.5
  • ...and 85 more