# Lazarsfeld-Mukai Reflexive Sheaves and their Stability

## TL;DR

This paper constructs and studies Lazarsfeld-Mukai reflexive sheaves on smooth projective varieties, proving their stability properties under specific conditions, thereby advancing understanding of their geometric and stability characteristics.

## Contribution

It introduces a new construction of Lazarsfeld-Mukai reflexive sheaves via elementary transformations and establishes their $ ext{mu}_L$-(semi)stability under certain conditions.

## Key findings

- Constructed reflexive sheaves via elementary transformations.
- Proved $ ext{mu}_L$-(semi)stability of these sheaves.
- Provided conditions ensuring stability properties.

## Abstract

Consider an ample and globally generated line bundle $L$ on a smooth projective variety $X$ of dimension $N\geq 2$ over $\mathbb{C}$. Let $D$ be a smooth divisor in the complete linear system of $L$. We construct reflexive sheaves on $X$ by an elementary transformation of a trivial bundle on $X$ along certain globally generated torsion-free sheaves on $D$. The dual reflexive sheaves are called the Lazarsfeld-Mukai reflexive sheaves. We prove the $\mu_L$-(semi)stability of such reflexive sheaves under certain conditions.