On the fixed locus of framed instanton sheaves on $\mathbb{P}^{3}$

Abdelmoubine Amar Henni

Vol. 308 (2020), No. 1, 41–72

DOI: 10.2140/pjm.2020.308.41

Abstract

Let $\mathbb{𝕋}$ be the three-dimensional torus acting on ${\mathbb{P}}^3$ and ${\mathcal{ℳ}}_{{\mathbb{P}}^3}^{\mathbb{𝕋}}\left(c\right)$ be the fixed locus of the corresponding action on the moduli space of rank $2$ framed instanton sheaves on ${\mathbb{P}}^3$. We prove that ${\mathcal{ℳ}}_{{\mathbb{P}}^3}^{\mathbb{𝕋}}\left(c\right)$ consist only of non-locally-free instanton sheaves whose double dual is the trivial bundle ${\mathcal{𝒪}}_{{\mathbb{P}}^3}^{\oplus 2}$. Moreover, we relate these instantons to Pandharipande–Thomas stable pairs and give a classification of their support. This allows us to compute a lower bound on the number of components of ${\mathcal{ℳ}}_{{\mathbb{P}}^3}^{\mathbb{𝕋}}\left(c\right)$.

Keywords

instantons, fixed locus, moduli spaces

Mathematical Subject Classification 2010

Primary: 14F05, 14J10

Milestones

Received: 17 August 2018

Revised: 10 October 2019

Accepted: 6 June 2020

Published: 3 December 2020