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Peer-reviewedCC BYGeometry & Topology

Counting surfaces on Calabi–Yau 4-folds, I : Foundations

  • Younghan Bae1
  • Martijn Kool2
  • Hyeonjun Park3
  1. 1Department of Mathematics, University of Michigan, Ann Arbor, MI, United States
  2. 2Mathematical Institute, Utrecht University, Utrecht, Netherlands
  3. 3June E Huh Center for Mathematical Challenges, Korea Institute for Advanced Study, Seoul, South Korea
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Abstract

This is the first in a series of papers on counting surfaces on Calabi-Yau 4-folds.To complement the Hilbert scheme of 2-dimensional subschemes, we introduce two types of moduli spaces of stable pairs.We show that all three moduli spaces are related by GIT wall-crossing and parametrize stable objects in the bounded derived category.We construct reduced Oh-Thomas virtual cycles on the moduli spaces via Kiem-Li cosection localization and prove that they are deformation invariant along Hodge loci.As an application, we show that the variational Hodge conjecture holds for any family of Calabi-Yau 4-folds supporting a nonzero reduced virtual cycle.1. Introduction 2431 2. Stable pairs 2443 3. Moduli spaces 2453 4. Virtual cycles 2481 5. Deformation invariance 2513 Appendix A. Reductions via .1/-shifted 1-forms 2529 Appendix B. Moduli stacks of semistable sheaves 2543 Appendix C. Orientations for relative moduli spaces 2554

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Cite

BibTeXRIS
BibTeX
@article{bae2026counting,
  title = {{Counting surfaces on Calabi–Yau 4-folds, I : Foundations}},
  author = {Younghan Bae and Martijn Kool and Hyeonjun Park},
  journal = {Geometry \& Topology},
  year = {2026},
  month = aug,
  doi = {10.2140/gt.2026.30.2431},
  url = {https://doi.org/10.2140/gt.2026.30.2431},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Counting surfaces on Calabi–Yau 4-folds, I : Foundations
AU  - Younghan Bae
AU  - Martijn Kool
AU  - Hyeonjun Park
PY  - 2026
DA  - 2026/08/13
JO  - Geometry & Topology
DO  - 10.2140/gt.2026.30.2431
UR  - https://doi.org/10.2140/gt.2026.30.2431
AB  - This is the first in a series of papers on counting surfaces on Calabi-Yau 4-folds.To complement the Hilbert scheme of 2-dimensional subschemes, we introduce two types of moduli spaces of stable pairs.We show that all three moduli spaces are related by GIT wall-crossing and parametrize stable objects in the bounded derived category.We construct reduced Oh-Thomas virtual cycles on the moduli spaces via Kiem-Li cosection localization and prove that they are deformation invariant along Hodge loci.As an application, we show that the variational Hodge conjecture holds for any family of Calabi-Yau 4-folds supporting a nonzero reduced virtual cycle.1. Introduction 2431 2. Stable pairs 2443 3. Moduli spaces 2453 4. Virtual cycles 2481 5. Deformation invariance 2513 Appendix A. Reductions via .1/-shifted 1-forms 2529 Appendix B. Moduli stacks of semistable sheaves 2543 Appendix C. Orientations for relative moduli spaces 2554
SN  - 1364-0380
SN  - 1465-3060
ER  -