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PreprintCC BY 4.0arXiv

A toric Calabi-Yau counterexample to Schimpf's connected stable pairs pole conjecture

  • Reginald Anderson
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Abstract

Schimpf's connected reformulation of Pandharipande's pole conjecture for stable pairs predicts the following restriction: in class β, connected descendent coefficients have nonzero p-poles only where −p is an m-th root of unity with 1≤ m≤divis(β). We give a toric counterexample. The threefold is a smooth quasi-projective toric Calabi--Yau threefold whose compact torus-invariant curves form a chain C1∪ C2. For the primitive class β=2[C1]+[C2], the coefficient of Qβz3 in the connected series for the insertion chz([v]T), where [v]T is the equivariant class of v=C1∩ C2, has Laurent expansion 9/41/(p−1)+O(1) after restricting to the one-parameter subtorus with tangent weights (1,2,−3) at v. Since divis(β)=1, Schimpf's condition allows only the nonzero pole p=−1. The pole at p=1 is therefore forbidden.

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Cite

BibTeXRIS
BibTeX
@misc{anderson2026toric,
  title = {{A toric Calabi-Yau counterexample to Schimpf's connected stable pairs pole conjecture}},
  author = {Reginald Anderson},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06876},
  eprint = {2610.06876},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06876},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A toric Calabi-Yau counterexample to Schimpf's connected stable pairs pole conjecture
AU  - Reginald Anderson
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.06876
UR  - https://arxiv.org/abs/2610.06876
AB  - Schimpf's connected reformulation of Pandharipande's pole conjecture for stable pairs predicts the following restriction: in class β, connected descendent coefficients have nonzero p-poles only where −p is an m-th root of unity with 1≤ m≤divis(β). We give a toric counterexample. The threefold is a smooth quasi-projective toric Calabi--Yau threefold whose compact torus-invariant curves form a chain C_1∪ C_2. For the primitive class β=2[C_1]+[C_2], the coefficient of Q^βz^3 in the connected series for the insertion ch_z([v]_T), where [v]_T is the equivariant class of v=C_1∩ C_2, has Laurent expansion 9/41/(p−1)+O(1) after restricting to the one-parameter subtorus with tangent weights (1,2,−3) at v. Since divis(β)=1, Schimpf's condition allows only the nonzero pole p=−1. The pole at p=1 is therefore forbidden.
ER  -