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The Moduli Space of Stable Coherent Sheaves via Non-archimedean Geometry

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Abstract

We provide a construction of the moduli space of stable coherent sheaves in the world of non-archimedean geometry, where we use the notion of Berkovich non-archimedean analytic spaces. The motivation for our construction is Tony Yue Yu’s non-archimedean enumerative geometry in Gromov—Witten theory. The construction of the moduli space of stable sheaves using Berkovich analytic spaces will give rise to the non-archimedean version of Donaldson—Thomas invariants. In this paper we give the moduli construction over a non-archimedean field \({\mathbb{K}}\). We use the machinery of formal schemes, that is, we define and construct the formal moduli stack of (semi)-stable coherent sheaves over a discrete valuation ring R, and taking generic fiber we get the non-archimedean analytic moduli space of semistable coherent sheaves over the fractional non-archimedean field \({\mathbb{K}}\). We generalize Joyce’s d-critical scheme structure in [37] or Kiem—Li’s virtual critical manifolds in [38] to the world of formal schemes, and Berkovich non-archimedean analytic spaces. As an application, we provide a proof for the motivic localization formula for a d-critical non-archimedean \({\mathbb{K}}\)-analytic space using global motive of vanishing cycles and motivic integration on oriented formal d-critical schemes. This generalizes Maulik’s motivic localization formula for the motivic Donaldson—Thomas invariants.

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Acknowledgements

The author would like to thank Yifeng Liu, Johannes Nicaise, and Tony Yue Yu for valuable discussions on Berkovich analytic spaces, especially Johannes Nicaise for answering questions about the motivic integration of formal schemes in [56], Yifeng Liu for the hospitality when visiting Northwestern University, and Tony Yue Yu for the correspondence on his non-archimedean enumerative geometry in Gromov—Witten theory and valuable suggestions on the generalized version of the motivic localization formula to non-archimedean analytic spaces.

Many thanks to Professors Tom Coates, Alessio Corti and R. Thomas for the support in Imperial College London, where the author started to think about the research along this direction. The author also thanks Professor Jun Li and Professor Dominic Joyce for the discussion of d-critical schemes and virtual critical manifolds, and Professor Sheldon Katz for the discussion about the motivic localization formula on motivic Donaldson—Thomas invariants when visiting UIUC in January 2017. The author thanks Professors Jim Bryan, Andrei Okounkov and Balazs Szendroi for email correspondence on the index formula of plane partitions, and especially thanks to Balazs Szendroi for pointing an error in the last example in an earlier version of the paper. This work is partially supported by NSF DMS-1600997.

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  1. Department of Mathematics, University of Kansas, 405 Snow Hall, 1460 Jayhawk Blvd, Lawrence, USA

    Yun Feng Jiang

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Correspondence to Yun Feng Jiang.

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Dedicated to Professor Banghe Li on His 80th Birthday

Partially supported by NSF (Grant No. DMS-1600997)

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Jiang, Y.F. The Moduli Space of Stable Coherent Sheaves via Non-archimedean Geometry. Acta. Math. Sin.-English Ser. 38, 1722–1780 (2022). https://doi.org/10.1007/s10114-022-2107-1

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