GW/PT-correspondence for CY $4$-fold-II: Virtual Classes

Abstract

Let B be a complex manifold and f:bX→ B a derived \textit{complex-analytic} (not necessarily \textit{algebraic}) space with an oriented relative (−2)-shifted symplectic structure. We suppose that the closed symplectic form is locally exact in the completed relative de Rham complex, and construct a derived smooth manifold over B, independent of additional choices. When f is proper and oriented, this gives integral virtual fundamental classes on its fibres. Our argument is heavily based on the strategy of Borisov-Joyce who already solved the \textit{algebraic} version of this problem, and uses \textit{derived differential geometry}. The proof also contains a relative analytic Darboux theorem a-la Brav-Bussi-Joyce, and has a natural analogue in Borel-Moore homology. We provide applications to complex-analytic enumerative geometry: in particular, to counting Pandharipande-Thomas pairs on compact and non-compact complex-analytic Calabi-Yau $4$-folds. This is part of a joint project with Felix Thimm dedicated to the proof of the GW/PT-correspondence for Calabi-Yau $4$-folds by implementing Pardon's programme in dimension $4$.

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BibTeX RIS

BibTeX

@misc{karpov2026gwptcorrespondence,
  title = {{GW/PT-correspondence for CY $4$-fold-II: Virtual Classes}},
  author = {Ivan Karpov},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.03566},
  eprint = {2611.03566},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.03566},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - GW/PT-correspondence for CY $4$-fold-II: Virtual Classes
AU  - Ivan Karpov
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.03566
UR  - https://arxiv.org/abs/2611.03566
AB  - Let B be a complex manifold and f:bX→ B a derived \textit{complex-analytic} (not necessarily \textit{algebraic}) space with an oriented relative (−2)-shifted symplectic structure. We suppose that the closed symplectic form is locally exact in the completed relative de Rham complex, and construct a derived smooth manifold over B, independent of additional choices. When f is proper and oriented, this gives integral virtual fundamental classes on its fibres. Our argument is heavily based on the strategy of Borisov-Joyce who already solved the \textit{algebraic} version of this problem, and uses \textit{derived differential geometry}. The proof also contains a relative analytic Darboux theorem a-la Brav-Bussi-Joyce, and has a natural analogue in Borel-Moore homology. We provide applications to complex-analytic enumerative geometry: in particular, to counting Pandharipande-Thomas pairs on compact and non-compact complex-analytic Calabi-Yau $4$-folds. This is part of a joint project with Felix Thimm dedicated to the proof of the GW/PT-correspondence for Calabi-Yau $4$-folds by implementing Pardon's programme in dimension $4$.
ER  -

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