Quadratic Donaldson-Thomas invariants for $(\mathbb{P}^1)^3$ and some other smooth proper toric threefolds
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| Format: | Preprint |
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2025
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| _version_ | 1866915242912514048 |
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| author | Levine, Marc Viergever, Anna M. |
| author_facet | Levine, Marc Viergever, Anna M. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_14420 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quadratic Donaldson-Thomas invariants for $(\mathbb{P}^1)^3$ and some other smooth proper toric threefolds Levine, Marc Viergever, Anna M. Algebraic Geometry Algebraic Topology 14N35, 14F42 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. |
| title | Quadratic Donaldson-Thomas invariants for $(\mathbb{P}^1)^3$ and some other smooth proper toric threefolds |
| topic | Algebraic Geometry Algebraic Topology 14N35, 14F42 |
| url | https://arxiv.org/abs/2503.14420 |