Counting higher-rank sheaves on toric threefolds
Abstract
We develop an equivariant K-theoretic vertex formalism for a colored, abelianized higher-rank sector on smooth projective toric threefolds X. Nonzero first Chern classes are encoded as magnetic fluxes and absorbed into line-bundle twists, after which the virtual character is redistributed into finite perturbative, vertex, and edge contributions. We compute the resulting (unreduced and trace-free) virtual dimensions and analyze the interaction terms relevant to a possible reduction to rank one. We emphasize that, on a compact toric threefold, colored partitions do not classify all torus-fixed stable higher-rank sheaves: the resulting series is therefore a proposed abelianized/framed vertex series unless an additional comparison theorem with the full stable-sheaf moduli problem is supplied. When the threefold obstruction theory is not available in the desired form, we pass to the local Calabi--Yau fourfold Y=Tot(KX) and formulate the corresponding colored DT4 vertex. A KX-twisted tautological insertion removes fixed points with nontrivial fiber layers, reducing solid partitions to plane partitions at the level of the local vertex. In rank one we give a scheme-theoretic explanation: the Hilbert functor of subschemes of X is canonically isomorphic to the constrained Hilbert functor of subschemes of Y lying scheme-theoretically in the fixed zero section. Under the hypotheses of the DT4 K-theoretic Lefschetz/virtual-pullback formalism, the associated support insertion reduces the rank-one fourfold theory to the threefold theory. We keep this global rank-one statement logically distinct from the higher-rank colored-vertex reduction.
Related
- Pointwise improvement of the error term in the higher rank hyperbolic lattice counting problem
- Decompositions of birational toric morphism between threefolds
- Decomposition of toric sheaves
- Deformations of toric pairs and the smoothing of Batyrev Calabi-Yau threefolds
- Mirror symmetry for gCICY threefolds (I): a [3,−1] block and a non-Gorenstein toric phase
BibTeX
@misc{piazzalunga2026counting,
title = {{Counting higher-rank sheaves on toric threefolds}},
author = {Nicolo Piazzalunga and Artan Sheshmani and Shing-Tung Yau},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04965},
eprint = {2610.04965},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04965},
}
RIS
TY - UNPB
TI - Counting higher-rank sheaves on toric threefolds
AU - Nicolo Piazzalunga
AU - Artan Sheshmani
AU - Shing-Tung Yau
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.04965
UR - https://arxiv.org/abs/2610.04965
AB - We develop an equivariant K-theoretic vertex formalism for a colored, abelianized higher-rank sector on smooth projective toric threefolds X. Nonzero first Chern classes are encoded as magnetic fluxes and absorbed into line-bundle twists, after which the virtual character is redistributed into finite perturbative, vertex, and edge contributions. We compute the resulting (unreduced and trace-free) virtual dimensions and analyze the interaction terms relevant to a possible reduction to rank one. We emphasize that, on a compact toric threefold, colored partitions do not classify all torus-fixed stable higher-rank sheaves: the resulting series is therefore a proposed abelianized/framed vertex series unless an additional comparison theorem with the full stable-sheaf moduli problem is supplied. When the threefold obstruction theory is not available in the desired form, we pass to the local Calabi--Yau fourfold Y=Tot(K_X) and formulate the corresponding colored DT4 vertex. A K_X-twisted tautological insertion removes fixed points with nontrivial fiber layers, reducing solid partitions to plane partitions at the level of the local vertex. In rank one we give a scheme-theoretic explanation: the Hilbert functor of subschemes of X is canonically isomorphic to the constrained Hilbert functor of subschemes of Y lying scheme-theoretically in the fixed zero section. Under the hypotheses of the DT4 K-theoretic Lefschetz/virtual-pullback formalism, the associated support insertion reduces the rank-one fourfold theory to the threefold theory. We keep this global rank-one statement logically distinct from the higher-rank colored-vertex reduction.
ER -