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arXiv · 2208.09474

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

Younghan Bae·Martijn Kool·Hyeonjun Park
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Abstract

This is the first part in a series of papers on counting surfaces on Calabi-Yau 4-folds. Besides the Hilbert scheme of 2-dimensional subschemes, we introduce \emph{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 \emph{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 non-zero reduced virtual cycle.

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Younghan Bae, Martijn Kool, Hyeonjun Park. 2022-08-19. Counting surfaces on Calabi-Yau 4-folds I: Foundations. https://doi.org/10.2140/gt.2026.30.2431

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