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Deformations of morphisms of sheaves

Donatella Iacono, Elena Martinengo

TL;DR

We address infinitesimal deformations of morphisms between locally free sheaves on a smooth projective variety by encoding the problem in differential graded Lie algebras (dgLas) and semicosimplicial methods. The main device is the Thom–Whitney totalization $ ext{Tot}({ rak g}^ abla)$ of suitably constructed semicosimplicial dgLas, which yields a Deligne functor equivalent to the deformation functor of the morphism; this covers both local and global cases and, via Hinich descent, provides a cover-independent description. Applied to coherent systems $( rak E,U)$, with $s:U o rak E$, we obtain a dgLa controlling deformations of the pair, and, under $H^1(X,\mathcal{O}_X)=0$ (and often $H^2(X,\mathcal{O}_X)=0$), derive a long exact sequence relating $H^i(X, ext{End} rak E)$, $ ext{Hom}(U,ullet)$, and the tangent/obstruction spaces; this yields Brill–Noether–type information over arbitrary char-zero algebraically closed fields. An explicit model for the governing semicosimplicial dgLa is provided, together with corollaries describing smoothness criteria and the influence on the deformation theory of coherent systems and sections. Overall, the paper furnishes a powerful dgLa-based toolkit to study deformations of morphisms of sheaves and their sections beyond the complex case, with clear cohomological consequences.

Abstract

We analyse infinitesimal deformations of morphisms of locally free sheaves on a smooth projective variety $X$ over an algebraically closed field of characteristic zero. In particular, we describe a differential graded Lie algebra controlling the deformation problem. As an application, we study infinitesimal deformations of pairs given by a locally free sheaf and a subspace of it sections with a view towards Brill-Noether theory.

Deformations of morphisms of sheaves

TL;DR

We address infinitesimal deformations of morphisms between locally free sheaves on a smooth projective variety by encoding the problem in differential graded Lie algebras (dgLas) and semicosimplicial methods. The main device is the Thom–Whitney totalization of suitably constructed semicosimplicial dgLas, which yields a Deligne functor equivalent to the deformation functor of the morphism; this covers both local and global cases and, via Hinich descent, provides a cover-independent description. Applied to coherent systems , with , we obtain a dgLa controlling deformations of the pair, and, under (and often ), derive a long exact sequence relating , , and the tangent/obstruction spaces; this yields Brill–Noether–type information over arbitrary char-zero algebraically closed fields. An explicit model for the governing semicosimplicial dgLa is provided, together with corollaries describing smoothness criteria and the influence on the deformation theory of coherent systems and sections. Overall, the paper furnishes a powerful dgLa-based toolkit to study deformations of morphisms of sheaves and their sections beyond the complex case, with clear cohomological consequences.

Abstract

We analyse infinitesimal deformations of morphisms of locally free sheaves on a smooth projective variety over an algebraically closed field of characteristic zero. In particular, we describe a differential graded Lie algebra controlling the deformation problem. As an application, we study infinitesimal deformations of pairs given by a locally free sheaf and a subspace of it sections with a view towards Brill-Noether theory.
Paper Structure (13 sections, 18 theorems, 90 equations)

This paper contains 13 sections, 18 theorems, 90 equations.

Table of Contents

  1. Preliminaries
  2. Differential graded Lie algebras, deformation and Deligne functors
  3. Semicosimplical differential graded Lie algebras
  4. Semicosimplicial groupoid and descent theorem
  5. Deformations of a morphism of locally free sheaves
  6. Geometric deformations
  7. The local case
  8. The global case
  9. Deformations of a locally free sheaf and a subspace of sections
  10. Geometric deformations.
  11. The dgLa that controls deformations of $(\mathcal{E},U)$
  12. An explicit model for the semicosimplicial dgla that controls deformations of $(\mathcal{E}, U)$.
  13. Deformations of a locally free sheaf preserving some sections

Key Result

Theorem 1

The functor $\mathop{\mathrm{Def}}\nolimits_{(\mathcal{F}, \alpha, \mathcal{G})}$ of infinitesimal deformations of the morphism $\alpha: \mathcal{F} \to \mathcal{G}$ is equivalent to the Deligne functor $\mathop{\mathrm{Del}}\nolimits_{\mathop{\mathrm{Tot}}\nolimits(H(\mathcal{V})^\Delta)}$ associat

Theorems & Definitions (60)

  • Theorem : Theorem \ref{['thm.Del eq Def']}
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Remark 1.7
  • Definition 1.8
  • Definition 1.9
  • ...and 50 more