A virtual $\mathrm{PGL}_r$-$\mathrm{SL}_r$ correspondence for projective surfaces
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910905845940224 |
|---|---|
| author | van Bree, D. Gholampour, A. Jiang, Y. Kool, M. |
| author_facet | van Bree, D. Gholampour, A. Jiang, Y. Kool, M. |
| contents | For a smooth projective surface $X$ satisfying $H_1(X,\mathbb{Z}) = 0$ and $w \in H^2(X,μ_r)$, we study deformation invariants of the pair $(X,w)$. Choosing a Brauer-Severi variety $Y$ (or, equivalently, Azumaya algebra $\mathcal{A}$) over $X$ with Stiefel-Whitney class $w$, the invariants are defined as virtual intersection numbers on suitable moduli spaces of stable twisted sheaves on $Y$ constructed by Yoshioka (or, equivalently, moduli spaces of $\mathcal{A}$-modules of Hoffmann-Stuhler).
We show that the invariants do not depend on the choice of $Y$. Using a result of de Jong, we observe that they are deformation invariants of the pair $(X,w)$. For surfaces with $h^{2,0}(X) > 0$, we show that the invariants can often be expressed as virtual intersection numbers on Gieseker-Maruyama-Simpson moduli spaces of stable sheaves on $X$. This can be seen as a $\mathrm{PGL}_r$-$\mathrm{SL}_r$ correspondence.
As an application, we express $\mathrm{SU}(r) / μ_r$ Vafa-Witten invariants of $X$ in terms of $\mathrm{SU}(r)$ Vafa-Witten invariants of $X$. We also show how formulae from Donaldson theory can be used to obtain upper bounds for the minimal second Chern class of Azumaya algebras on $X$ with given division algebra at the generic point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_02288 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A virtual $\mathrm{PGL}_r$-$\mathrm{SL}_r$ correspondence for projective surfaces van Bree, D. Gholampour, A. Jiang, Y. Kool, M. Algebraic Geometry High Energy Physics - Theory Differential Geometry 14D20, 14D21, 14J60 (Primary), 14F22, 14J80 (Secondary) For a smooth projective surface $X$ satisfying $H_1(X,\mathbb{Z}) = 0$ and $w \in H^2(X,μ_r)$, we study deformation invariants of the pair $(X,w)$. Choosing a Brauer-Severi variety $Y$ (or, equivalently, Azumaya algebra $\mathcal{A}$) over $X$ with Stiefel-Whitney class $w$, the invariants are defined as virtual intersection numbers on suitable moduli spaces of stable twisted sheaves on $Y$ constructed by Yoshioka (or, equivalently, moduli spaces of $\mathcal{A}$-modules of Hoffmann-Stuhler). We show that the invariants do not depend on the choice of $Y$. Using a result of de Jong, we observe that they are deformation invariants of the pair $(X,w)$. For surfaces with $h^{2,0}(X) > 0$, we show that the invariants can often be expressed as virtual intersection numbers on Gieseker-Maruyama-Simpson moduli spaces of stable sheaves on $X$. This can be seen as a $\mathrm{PGL}_r$-$\mathrm{SL}_r$ correspondence. As an application, we express $\mathrm{SU}(r) / μ_r$ Vafa-Witten invariants of $X$ in terms of $\mathrm{SU}(r)$ Vafa-Witten invariants of $X$. We also show how formulae from Donaldson theory can be used to obtain upper bounds for the minimal second Chern class of Azumaya algebras on $X$ with given division algebra at the generic point. |
| title | A virtual $\mathrm{PGL}_r$-$\mathrm{SL}_r$ correspondence for projective surfaces |
| topic | Algebraic Geometry High Energy Physics - Theory Differential Geometry 14D20, 14D21, 14J60 (Primary), 14F22, 14J80 (Secondary) |
| url | https://arxiv.org/abs/2308.02288 |