Quadratic Donaldson-Thomas invariants for $(\mathbb{P}^1)^3$ and some other smooth proper toric threefolds

Marc Levine; Anna M. Viergever

doi:10.48550/arXiv.2503.14420

Quadratic Donaldson-Thomas invariants for $(\mathbb{P}^1)^3$ and some other smooth proper toric threefolds

Marc Levine
, Anna M. Viergever

Institute of Algebraic Geometry

Universität Duisburg-Essen

Research output: Working paper/Preprint › Preprint

Abstract

Using virtual localization in Witt sheaf cohomology, we show that the generating series of quadratic Donaldson-Thomas invariants of $(\mathbb{P}^1)^3$, valued in the Witt ring of $\mathbb{R}$, $W(\mathbb{R})\cong \mathbb{Z}$, is equal to $M(q^2)^{-8}$, where $M(q)$ is the MacMahon function. This confirms a modified version of a conjecture of Viergever. We also show that a localized version of this conjecture holds for certain iterated blow-ups of $(\mathbb{P}^1)^3$ and other related smooth proper toric varieties.

Original language	English
DOIs	https://doi.org/10.48550/arXiv.2503.14420
Publication status	E-pub ahead of print - 18 Mar 2025

Keywords

math.AG
math.AT

Access to Document

10.48550/arXiv.2503.14420Licence: CC BY 4.0

Cite this

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title = "Quadratic Donaldson-Thomas invariants for \$(\textbackslash{}mathbb\{P\}\textasciicircum{}1)\textasciicircum{}3\$ and some other smooth proper toric threefolds",

abstract = "Using virtual localization in Witt sheaf cohomology, we show that the generating series of quadratic Donaldson-Thomas invariants of \$(\textbackslash{}mathbb\{P\}\textasciicircum{}1)\textasciicircum{}3\$, valued in the Witt ring of \$\textbackslash{}mathbb\{R\}\$, \$W(\textbackslash{}mathbb\{R\})\textbackslash{}cong \textbackslash{}mathbb\{Z\}\$, is equal to \$M(q\textasciicircum{}2)\textasciicircum{}\{-8\}\$, where \$M(q)\$ is the MacMahon function. This confirms a modified version of a conjecture of Viergever. We also show that a localized version of this conjecture holds for certain iterated blow-ups of \$(\textbackslash{}mathbb\{P\}\textasciicircum{}1)\textasciicircum{}3\$ and other related smooth proper toric varieties.",

keywords = "math.AG, math.AT",

author = "Marc Levine and Viergever, \{Anna M.\}",

note = "In the introduction, we altered our main conjecture slightly, deleting a condition for the precise value of the constant \$ε\textbackslash{}in\textbackslash{}\{\textbackslash{}pm1\textbackslash{}\}\$, and added comments about the vanishing of the odd quadratic DT invariants",

year = "2025",

month = mar,

day = "18",

doi = "10.48550/arXiv.2503.14420",

language = "English",

type = "WorkingPaper",

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T1 - Quadratic Donaldson-Thomas invariants for $(\mathbb{P}^1)^3$ and some other smooth proper toric threefolds

AU - Levine, Marc

AU - Viergever, Anna M.

N1 - In the introduction, we altered our main conjecture slightly, deleting a condition for the precise value of the constant $ε\in\{\pm1\}$, and added comments about the vanishing of the odd quadratic DT invariants

PY - 2025/3/18

Y1 - 2025/3/18

N2 - Using virtual localization in Witt sheaf cohomology, we show that the generating series of quadratic Donaldson-Thomas invariants of $(\mathbb{P}^1)^3$, valued in the Witt ring of $\mathbb{R}$, $W(\mathbb{R})\cong \mathbb{Z}$, is equal to $M(q^2)^{-8}$, where $M(q)$ is the MacMahon function. This confirms a modified version of a conjecture of Viergever. We also show that a localized version of this conjecture holds for certain iterated blow-ups of $(\mathbb{P}^1)^3$ and other related smooth proper toric varieties.

AB - Using virtual localization in Witt sheaf cohomology, we show that the generating series of quadratic Donaldson-Thomas invariants of $(\mathbb{P}^1)^3$, valued in the Witt ring of $\mathbb{R}$, $W(\mathbb{R})\cong \mathbb{Z}$, is equal to $M(q^2)^{-8}$, where $M(q)$ is the MacMahon function. This confirms a modified version of a conjecture of Viergever. We also show that a localized version of this conjecture holds for certain iterated blow-ups of $(\mathbb{P}^1)^3$ and other related smooth proper toric varieties.

KW - math.AG

KW - math.AT

U2 - 10.48550/arXiv.2503.14420

DO - 10.48550/arXiv.2503.14420

M3 - Preprint

BT - Quadratic Donaldson-Thomas invariants for $(\mathbb{P}^1)^3$ and some other smooth proper toric threefolds

ER -