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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2409.17358 |
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| _version_ | 1866912427701960704 |
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| author | Groechenig, Michael Wyss, Dimitri Ziegler, Paul |
| author_facet | Groechenig, Michael Wyss, Dimitri Ziegler, Paul |
| contents | We study $p$-adic manifolds associated with twisted points of quotient stacks $\mathcal{X} = [U/G]$ and their quotient spaces $π:\mathcal{X} \to X$. We prove several structural results about the fibres of $π$ and derive in particular a formula expressing $p$-adic integrals on $X$ in terms of the cyclotomic inertia stack of $\mathcal{X}$, generalizing the orbifold formula for Deligne-Mumford stacks.
We then apply our formalism to moduli problems associated to hereditary abelian categories with symmetric Euler pairing, and show that their refined BPS-invariants are computed locally on the coarse moduli space by a $p$-adic integral. As a consequence we recover the $χ$-independence of these invariants for $1$-dimensional sheaves on del Pezzo surfaces previously proven by Maulik--Shen. Along the way we derive a new formula for the plethystic logarithm on the $λ$-ring of functions on $k$-linear stacks, which might be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_17358 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Twisted points of quotient stacks, integration and BPS-invariants Groechenig, Michael Wyss, Dimitri Ziegler, Paul Algebraic Geometry We study $p$-adic manifolds associated with twisted points of quotient stacks $\mathcal{X} = [U/G]$ and their quotient spaces $π:\mathcal{X} \to X$. We prove several structural results about the fibres of $π$ and derive in particular a formula expressing $p$-adic integrals on $X$ in terms of the cyclotomic inertia stack of $\mathcal{X}$, generalizing the orbifold formula for Deligne-Mumford stacks. We then apply our formalism to moduli problems associated to hereditary abelian categories with symmetric Euler pairing, and show that their refined BPS-invariants are computed locally on the coarse moduli space by a $p$-adic integral. As a consequence we recover the $χ$-independence of these invariants for $1$-dimensional sheaves on del Pezzo surfaces previously proven by Maulik--Shen. Along the way we derive a new formula for the plethystic logarithm on the $λ$-ring of functions on $k$-linear stacks, which might be of independent interest. |
| title | Twisted points of quotient stacks, integration and BPS-invariants |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2409.17358 |