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Counting surfaces on
Calabi–Yau $4$-folds, I: Foundations
Younghan Bae, Martijn Kool and Hyeonjun Park |
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Geometry & Topology 30 (2026) 2431–2564
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Abstract |
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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. |
Keywords
Donaldson–Thomas theory, Calabi–Yau 4-folds, surfaces,
Hodge theory, derived algebraic geometry, geometric
invariant theory
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Mathematical Subject Classification
Primary: 14N35
Secondary: 14D07, 14F08, 14J35, 14J60, 14L24
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References |
Publication
Received: 13 November 2023
Revised: 15 January 2025
Accepted: 4 March 2025
Published: 13 August 2026
Proposed: Jim Bryan
Seconded: Marc Levine, Dan Abramovich
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Authors |
| Younghan Bae | |
| Department of Mathematics University of Michigan Ann Arbor, MI United States |
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| Martijn Kool | |
| Mathematical Institute Utrecht University Utrecht Netherlands |
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| Hyeonjun Park | |
| June E Huh Center for Mathematical
Challenges Korea Institute for Advanced Study Seoul South Korea |
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| © 2026 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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