#### Vol. 308, No. 1, 2020

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

### Abdelmoubine Amar Henni

Vol. 308 (2020), No. 1, 41–72
##### Abstract

Let $\mathbb{𝕋}$ be the three-dimensional torus acting on ${ℙ}^{3}$ and ${\mathsc{ℳ}}_{{ℙ}^{3}}^{\mathbb{𝕋}}\left(c\right)$ be the fixed locus of the corresponding action on the moduli space of rank $2$ framed instanton sheaves on ${ℙ}^{3}$. We prove that ${\mathsc{ℳ}}_{{ℙ}^{3}}^{\mathbb{𝕋}}\left(c\right)$ consist only of non-locally-free instanton sheaves whose double dual is the trivial bundle ${\mathsc{𝒪}}_{{ℙ}^{3}}^{\oplus 2}$. 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 ${\mathsc{ℳ}}_{{ℙ}^{3}}^{\mathbb{𝕋}}\left(c\right)$.

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##### 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
##### Authors
 Abdelmoubine Amar Henni Departamento de Matemática Universidade Federal de Santa Catarina Florianópolis, SC Brazil