Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generalized Brieskorn Modules III. The algebra $\tilde{\mathcal{A}}\_{conv.}$

Daniel Barlet

TL;DR

The paper develops a comprehensive convergent (a,b)-algebra, $\tilde{\mathcal{A}}_{conv.}$, housing both $a$ and $b$ with $ab-ba=b^2$ and acting on holomorphic germs in $a$. It proves that geometric convergent (a,b)-modules are naturally left $\tilde{\mathcal{A}}_{conv.}$-modules, free of finite type over $B$, and that morphisms linear over $B[a]$ extend to $\tilde{\mathcal{A}}_{conv.}$-linear morphisms via a Division Theorem. The work constructs a functorial complex of sheaves of left $\tilde{\mathcal{A}}_{conv.}$-modules on the zero set of a holomorphic function, linking the cohomology to Gauss-Manin connections and recovering generalized Brieskorn modules modulo $b$-torsion; these modules encapsulate nearby and vanishing cycle information in a convergent framework. A global finiteness theorem demonstrates that, under properness assumptions, global GBMs are small, with $b$-torsion finite and associated GBMs geometric, while raising questions about the interplay with mixed Hodge structures and Bernstein spectra. Overall, the paper provides a robust algebraic-analytic scaffold to relate convergent $(a,b)$-module structures to the topology of singular fibers and their Hodge-theoretic content.

Abstract

In this paper we introduce and study the ''convergent'' algebra (containing ''a'' and ''b'' and acting on holomorphic germs in ''a'') which naturally acts on the ''generalized Brieskorn modules'' associated to the Gauss-Manin connections of the germs at each point of the singular set of a holomorphic function on a complex manifold. We generalize to this convergent setting the results previously obtained (see 8], [9], [15] and [16]) in the formal case, and we show that, in suitable global situations (for instance when f is projective) we obtain also generalized (geometric) Brieskorn modules. So the question of the relationship between the left module structure on this algebra (which defines several interesting filtrations) and the mixte Hodge structure is raised

Generalized Brieskorn Modules III. The algebra $\tilde{\mathcal{A}}\_{conv.}$

TL;DR

The paper develops a comprehensive convergent (a,b)-algebra, , housing both and with and acting on holomorphic germs in . It proves that geometric convergent (a,b)-modules are naturally left -modules, free of finite type over , and that morphisms linear over extend to -linear morphisms via a Division Theorem. The work constructs a functorial complex of sheaves of left -modules on the zero set of a holomorphic function, linking the cohomology to Gauss-Manin connections and recovering generalized Brieskorn modules modulo -torsion; these modules encapsulate nearby and vanishing cycle information in a convergent framework. A global finiteness theorem demonstrates that, under properness assumptions, global GBMs are small, with -torsion finite and associated GBMs geometric, while raising questions about the interplay with mixed Hodge structures and Bernstein spectra. Overall, the paper provides a robust algebraic-analytic scaffold to relate convergent -module structures to the topology of singular fibers and their Hodge-theoretic content.

Abstract

In this paper we introduce and study the ''convergent'' algebra (containing ''a'' and ''b'' and acting on holomorphic germs in ''a'') which naturally acts on the ''generalized Brieskorn modules'' associated to the Gauss-Manin connections of the germs at each point of the singular set of a holomorphic function on a complex manifold. We generalize to this convergent setting the results previously obtained (see 8], [9], [15] and [16]) in the formal case, and we show that, in suitable global situations (for instance when f is projective) we obtain also generalized (geometric) Brieskorn modules. So the question of the relationship between the left module structure on this algebra (which defines several interesting filtrations) and the mixte Hodge structure is raised
Paper Structure (61 sections, 25 theorems, 164 equations)

This paper contains 61 sections, 25 theorems, 164 equations.

Table of Contents

  1. Abstract.
  2. AMS classification.
  3. Key words.
  4. The algebra $\tilde{\mathcal{A}}_{conv.}$
  5. First properties
  6. Notations.
  7. Two easy facts.
  8. proof.
  9. Remark.
  10. proof.
  11. An example.
  12. Exercise.
  13. Proof.
  14. Remark.
  15. Remark.
  16. ...and 46 more sections

Key Result

Lemma 1.1.1

We have the following recursion relations:

Theorems & Definitions (31)

  • Lemma 1.1.1
  • Corollary 1.1.2
  • Proposition 1.1.3
  • Corollary 1.1.4
  • Lemma 1.1.5
  • Lemma 1.1.6
  • Proposition 1.1.7
  • Lemma 1.1.8
  • Corollary 1.1.9
  • Proposition 1.2.1
  • ...and 21 more