Codimension one distributions and stable rank 2 reflexive sheaves on threefolds.

Omegar Calvo-Andrade,Marcos Jardim,Maurício Corrêa

doi:10.1590/0001-3765202120190909

Abstract

% We show that codimension one distributions with at most isolated singularities on certain smooth projective threefolds with Picard rank one have stable tangent sheaves. The ideas in the proof of this fact are then applied to the characterization of certain irreducible components of the moduli space of stable rank 2 reflexive sheaves on $\p3$, and to the construction of stable rank 2 reflexive sheaves with prescribed Chern classes on general threefolds. We also prove that if $\sG$ is a subfoliation of a codimension one distribution $\sF$ with isolated singularities, then $\sing(\sG)$ is a curve. As a consequence, we give a criterion to decide whether $\sG$ is globally given as the intersection of $\sF$ with another codimension one distribution. Turning our attention to codimension one distributions with non isolated singularities, we determine the number of connected components of the pure 1-dimensional component of the singular scheme.

Highlights

A codimension r distribution F on a smooth complex manifold X is given by an exact sequence
We prove that if G is a subfoliation of a distribution F with isolated singularities, G has non isolated singularities
The case d = 1 was considered by (Chang 1984, Theorem 3.14), who showed that the component described in Theorem 3 is the whole moduli space of stable rank 2 reflexive sheaves with Chern classes (c1, c2, c3) = (–1, 3, 5); see (Calvo-Andrade et al 2018, Theorem 8.1)

Summary

INTRODUCTION

A codimension r distribution F on a smooth complex manifold X is given by an exact sequence. Generic codimension one distributions on projective threefolds with rank one Picard group cannot be integrable. The case d = 1 was considered by (Chang 1984, Theorem 3.14), who showed that the component described in Theorem 3 is the whole moduli space of stable rank 2 reflexive sheaves with Chern classes (c1, c2, c3) = (–1, 3, 5); see (Calvo-Andrade et al 2018, Theorem 8.1). The existence of μ-stable reflexive sheaves with prescribed Chern classes is an open problem with particular interest to String Theory when X is a Calabi–Yau threefold In this context, we prove the following existence and uniqueness result for rank two reflexive sheaves; set γX := min{t ∈ Z | Ω1X(t) is globally generated}. The results listed above are subsequently proved in the five sections that follow

GENERIC DISTRIBUTIONS ON THREEFOLDS

CONNECTEDNESS OF NON ISOLATED SINGULARITIES

The standard exact sequence

Note that