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Bernstein-Sato polynomials for projective hypersurfaces with weighted homogeneous isolated singularities

Morihiko Saito

TL;DR

This work develops a computationally efficient framework to determine the roots of the Bernstein-Sato polynomial for projective hypersurfaces with weighted homogeneous isolated singularities by proving $E_2$-degeneration of the pole order spectral sequence under (IS) and (WH). The method reduces root calculation to the Hilbert series of the Jacobian ring except when $f$ is annihilated by a degree-zero vector field, in which case the analysis incorporates dualities via vanishing cycles, Brieskorn modules, and twisted de Rham complexes. In dimension three with $d>4$, the roots are described as $\tfrac{1}{d}(\mathbb{Z}\cap[3,k']) \cup R_Z$ away from the extremely degenerated case, with $k'$, $\nu_k$, and local Milnor data governing the structure. The approach integrates microlocal Gauss-Manin theory, pole order filtrations, and explicit spectral computations, yielding both general theorems and concrete examples that illustrate the interplay between global and local data in Bernstein-Sato theory.

Abstract

We present a quite efficient method to calculate the roots of Bernstein-Sato polynomial for a defining polynomial $f$ of a projective hypersurface $Z\subset{\mathbb P}^{n-1}$ of degree $d$ having only weighted homogeneous isolated singularities. We prove the $E_2$-degeneration of the pole order spectral sequence so that the computation of roots is reduced to the one of the Hilbert series of the Jacobian ring of $f$ except the special case where $f$ is annihilated by a nonzero vector field on ${\mathbb C}^n$ with linear function coefficients. In the three variable case with $d>4$ we may assume that this vector field is a linear combination of $x\partial_x, y\partial_y, z\partial_z$, where $f$ is called extremely degenerated; in particular, the latter case does not contain any essential indecomposable central hyperplane arrangement in ${\mathbb C}^3$. Combined with the self-duality of the Koszul complex and a theorem of Dimca and Popescu, it implies for $n=3$ with $d>4$ except the extremely degenerated case that $R_f=\frac{1}{d}({\mathbb Z}\cap[3,k'])\cup R_Z$. Here $R_f,R_Z$ are the roots of Bernstein-Sato polynomials of $f$ and $Z$ up to sign, and $k'=\max(2d-3,k_{\max}+3)$ with $k_{\max}$ the maximal degree of the ``torsion part" of the Jacobian ring, where the latter is known to be at most $2d-5$ in the hyperplane arrangement case.

Bernstein-Sato polynomials for projective hypersurfaces with weighted homogeneous isolated singularities

TL;DR

This work develops a computationally efficient framework to determine the roots of the Bernstein-Sato polynomial for projective hypersurfaces with weighted homogeneous isolated singularities by proving -degeneration of the pole order spectral sequence under (IS) and (WH). The method reduces root calculation to the Hilbert series of the Jacobian ring except when is annihilated by a degree-zero vector field, in which case the analysis incorporates dualities via vanishing cycles, Brieskorn modules, and twisted de Rham complexes. In dimension three with , the roots are described as away from the extremely degenerated case, with , , and local Milnor data governing the structure. The approach integrates microlocal Gauss-Manin theory, pole order filtrations, and explicit spectral computations, yielding both general theorems and concrete examples that illustrate the interplay between global and local data in Bernstein-Sato theory.

Abstract

We present a quite efficient method to calculate the roots of Bernstein-Sato polynomial for a defining polynomial of a projective hypersurface of degree having only weighted homogeneous isolated singularities. We prove the -degeneration of the pole order spectral sequence so that the computation of roots is reduced to the one of the Hilbert series of the Jacobian ring of except the special case where is annihilated by a nonzero vector field on with linear function coefficients. In the three variable case with we may assume that this vector field is a linear combination of , where is called extremely degenerated; in particular, the latter case does not contain any essential indecomposable central hyperplane arrangement in . Combined with the self-duality of the Koszul complex and a theorem of Dimca and Popescu, it implies for with except the extremely degenerated case that . Here are the roots of Bernstein-Sato polynomials of and up to sign, and with the maximal degree of the ``torsion part" of the Jacobian ring, where the latter is known to be at most in the hyperplane arrangement case.

Paper Structure

This paper contains 34 sections, 17 theorems, 255 equations.

Table of Contents

  1. Preliminaries
  2. Algebraic microlocal Gauss-Manin systems
  3. Pole order spectral sequences
  4. Isolated singularity case
  5. Weighted homogeneous isolated singularity case
  6. Compatibility of duality isomorphisms
  7. Proof of Proposition \ref{['P1']}
  8. Steenbrink spectrum and Bernstein-Sato polynomials
  9. Special hypersurfaces
  10. Fundamental exact sequences of vanishing cycles
  11. Calculation of the Milnor fiber cohomology
  12. Key theorem to the proof of Theorem \ref{['T2']}
  13. Proof of Theorem \ref{['T10.1']}, Part I
  14. Proof of Theorem \ref{['T10.1']}, Part II
  15. Proof of the exact sequence (\ref{['12.10']})
  16. ...and 19 more sections

Key Result

Theorem 1

Assume $\alpha\notin{\mathcal{R}}_Z$. If $\alpha$ satisfies the condition then If condition 2 does not hold, then only the implication $\Longleftarrow$ holds in 3.

Theorems & Definitions (46)

  • Theorem 1: bcm
  • Remark 1
  • Proposition 1
  • Theorem 2
  • Corollary 1
  • Theorem 3
  • Corollary 3
  • Remark 2
  • Theorem 4
  • Corollary 4
  • ...and 36 more