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Sur certains faisceaux de formes méromorphes en géométrie analytique complexe II

Kaddar Mohamed

TL;DR

This work develops a comprehensive framework for the functorial behavior of the sheaves ${\mathcal{L}}^{\bullet}_{X}$ of meromorphic forms that lift holomorphically to every desingularization of a reduced complex space $X$ of pure dimension $m$. It constructs natural pullback and higher direct image morphisms under broad classes of morphisms, using two complementary approaches: fiber integration with relative Kähler forms and duality/trace arguments on dualizing complexes. The paper proves independence from auxiliary choices (desingularizations, Kähler forms) and establishes base-change and composition compatibilities, unifying and extending classical results such as Grauert–Riemenschneider vanishing and the theory of singularities. By addressing both restriction and fiber-wise integration, it provides analytic tools for studying $L^{2}$-type meromorphic forms on singular spaces and their functorial properties in families.

Abstract

We study some functorial properties of certain sheaves of meromorphic forms on reduced complex space; particulary, the meromorphic forms which extend holomorphicaly on any desingularisation. The purpose concern their behavior under pull back and higher direct image (and in some case by integration on fibres of equidimensional or open morphism).

Sur certains faisceaux de formes méromorphes en géométrie analytique complexe II

TL;DR

This work develops a comprehensive framework for the functorial behavior of the sheaves of meromorphic forms that lift holomorphically to every desingularization of a reduced complex space of pure dimension . It constructs natural pullback and higher direct image morphisms under broad classes of morphisms, using two complementary approaches: fiber integration with relative Kähler forms and duality/trace arguments on dualizing complexes. The paper proves independence from auxiliary choices (desingularizations, Kähler forms) and establishes base-change and composition compatibilities, unifying and extending classical results such as Grauert–Riemenschneider vanishing and the theory of singularities. By addressing both restriction and fiber-wise integration, it provides analytic tools for studying -type meromorphic forms on singular spaces and their functorial properties in families.

Abstract

We study some functorial properties of certain sheaves of meromorphic forms on reduced complex space; particulary, the meromorphic forms which extend holomorphicaly on any desingularisation. The purpose concern their behavior under pull back and higher direct image (and in some case by integration on fibres of equidimensional or open morphism).

Paper Structure

This paper contains 21 sections, 11 theorems, 216 equations.

Table of Contents

  1. Cactérisations classiques .
  2. Formes de première espèce.
  3. Caractérisations.
  4. Quelques résultats basiques.
  5. Propriétés fondamentales.
  6. Image réciproque pour certaines classes de morphismes.
  7. Remarques.
  8. Morphisme de restriction: cas particulier.
  9. La restriction sur les faisceaux ${\mathcal{L}}^{\bullet}_{X}$.
  10. Construction et définition d'une restriction dans le cas lisse.
  11. Morphisme de restriction: cas général
  12. La preuve du \ref{['T4']}.
  13. Construction et définition de la restriction.
  14. La preuve du \ref{['T1']}.
  15. Image directe supérieure pour un morphisme géométriquement plat et ses conséquences.
  16. ...and 6 more sections

Key Result

Proposition 1.1

(Picard [P] tome 1 p.178) Soit $(X,0)$ un germe d'hypersurface réduite d' équation $f(x_{1},\cdots, x_{m+1}) = 0$ dans $({\Bbb C}^{m+1},0)$ et $\Sigma$ son lieu singulier (ou un sous ensemble rare dans $X$). Alors: (i) toute $m$ - forme méromorphe $\xi$ sur $X$ s'écrit avec $\varphi$ méromorphe sur $X$. (ii) pour que $\xi$ reste fini au voisinage de $O$, il faut que a) $\varphi$ soit holomorphe a

Theorems & Definitions (22)

  • Proposition 1.1
  • Proposition 1.2
  • Proposition 1.8
  • Proposition 1.9
  • proof
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • proof
  • ...and 12 more