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Sur certains faisceaux de formes méromorphes sur un espace complexe réduit

Mohamed Kaddar

TL;DR

The paper develops a comprehensive framework for meromorphic regular forms on reduced complex spaces, anchoring the theory in Grothendieck duality and the trace property. It introduces and compares several sheaves of meromorphic forms ($oldsymbol{ ilde{oldsymbol u}}$, $oldsymbol{aroldsymbol abla}$, $oldsymbol{ m L}$, $oldsymbol{ u}$) and establishes pullback and pushforward operations that are compatible with base-change, desingularization, and Nash modification. A central achievement is the construction of canonical trace morphisms for equidimensional and geometrically flat morphisms, linking relative regular forms to the relative canonical sheaf via duality (Kleiman- and RRV-type frameworks). The results unify absolute and relative theories, show stability and failure modes under various geometric operations, and provide explicit examples and corollaries that illuminate how normality/Cohen–Macaulay conditions influence duality and trace phenomena in singular settings.

Abstract

The purpose of this is the study of certain coherent sheaves of meromorphic forms on reduced complex space and particularly their behavior with respect to pull back and higher direct image.

Sur certains faisceaux de formes méromorphes sur un espace complexe réduit

TL;DR

The paper develops a comprehensive framework for meromorphic regular forms on reduced complex spaces, anchoring the theory in Grothendieck duality and the trace property. It introduces and compares several sheaves of meromorphic forms (, , , ) and establishes pullback and pushforward operations that are compatible with base-change, desingularization, and Nash modification. A central achievement is the construction of canonical trace morphisms for equidimensional and geometrically flat morphisms, linking relative regular forms to the relative canonical sheaf via duality (Kleiman- and RRV-type frameworks). The results unify absolute and relative theories, show stability and failure modes under various geometric operations, and provide explicit examples and corollaries that illuminate how normality/Cohen–Macaulay conditions influence duality and trace phenomena in singular settings.

Abstract

The purpose of this is the study of certain coherent sheaves of meromorphic forms on reduced complex space and particularly their behavior with respect to pull back and higher direct image.
Paper Structure (43 sections, 15 theorems, 398 equations)

This paper contains 43 sections, 15 theorems, 398 equations.

Table of Contents

  1. Formes méromorphes régulières absolues.
  2. Préliminaires.
  3. Formes holomorphes et torsion.
  4. Module dualisant.
  5. Faisceaux dualisants de Andréotti-Kas-Golovin.
  6. Formes méromorphes régulières et propriété de la trace.
  7. Formes méromorphes régulières et faisceau ${\omega}^{k}_{Z}$.
  8. Formes méromorphes régulières et courants.
  9. Notations et définitions.
  10. Propriétés fondamentales et résultats basiques.
  11. Les propriétés.
  12. Remarques et quelques exemples.
  13. Comportement par image réciproque et restriction.
  14. Comportement par cup produit.
  15. le faisceau ${\mathcal{L}}^{\bullet}_{X}$.
  16. ...and 28 more sections

Key Result

Proposition 2.1

Soit $X$ un espace analytique complexe réduit et de dimension pure $m$. Alors, pour tout entier $k\geq 0$, on a (i)$\omega^{k}_{X}$ est de profondeur au moins $2$; en particulier, pour tout sous espace $T$ de codimension 2 dans $X$, on a l'annulation des faisceaux de cohomologie ${\mathcal{H}}^{0}_{ injectif dans le cas d'une modification propre et surjectif dans le cas d 'un morphisme fini et ouv

Theorems & Definitions (34)

  • Proposition 2.1
  • proof
  • Proposition 5.1
  • Proposition 5.2
  • Proposition 5.3
  • proof
  • Proposition 5.4
  • proof
  • Proposition 6.1
  • proof
  • ...and 24 more