On rigid varieties with projective reduction
Author:
Shizhang Li
Journal:
J. Algebraic Geom. 29 (2020), 669-690
DOI:
https://doi.org/10.1090/jag/740
Published electronically:
November 4, 2019
MathSciNet review:
4158462
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Abstract |
References |
Additional Information
Abstract: In this paper, we study smooth proper rigid varieties which admit formal models whose special fibers are projective. The Main Theorem asserts that the identity components of the associated rigid Picard varieties will automatically be proper. Consequently, we prove that $p$-adic Hopf varieties will never have a projective reduction. The proof of our Main Theorem uses the theory of moduli of semistable coherent sheaves.
References
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- Werner Lütkebohmert, Formal-algebraic and rigid-analytic geometry, Math. Ann. 286 (1990), no. 1-3, 341–371. MR 1032938, DOI https://doi.org/10.1007/BF01453580
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References
- Siegfried Bosch and Werner Lütkebohmert, Formal and rigid geometry. I. Rigid spaces, Math. Ann. 295 (1993), no. 2, 291–317. MR 1202394, DOI https://doi.org/10.1007/BF01444889
- Siegfried Bosch and Werner Lütkebohmert, Formal and rigid geometry. II. Flattening techniques, Math. Ann. 296 (1993), no. 3, 403–429. MR 1225983, DOI https://doi.org/10.1007/BF01445112
- Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Formal and rigid geometry. III. The relative maximum principle, Math. Ann. 302 (1995), no. 1, 1–29. MR 1329445, DOI https://doi.org/10.1007/BF01444485
- Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Formal and rigid geometry. IV. The reduced fibre theorem, Invent. Math. 119 (1995), no. 2, 361–398. MR 1312505, DOI https://doi.org/10.1007/BF01245187
- Sarah Glaz, Commutative coherent rings, Lecture Notes in Mathematics, vol. 1371, Springer-Verlag, Berlin, 1989. MR 999133
- Urs Hartl and Werner Lütkebohmert, On rigid-analytic Picard varieties, J. Reine Angew. Math. 528 (2000), 101–148. MR 1801659, DOI https://doi.org/10.1515/crll.2000.087
- Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010. MR 2665168
- D. Hansen and S. Li, Line bundles on rigid varieties and Hodge symmetry, arXiv e-prints, 2017.
- Roland Huber, Étale cohomology of rigid analytic varieties and adic spaces, Aspects of Mathematics, E30, Friedr. Vieweg & Sohn, Braunschweig, 1996. MR 1734903
- Kiran S. Kedlaya and Ruochuan Liu, Relative $p$-adic Hodge theory: foundations, Astérisque 371 (2015), 239 (English, with English and French summaries). MR 3379653
- Finn Faye Knudsen and David Mumford, The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”, Math. Scand. 39 (1976), no. 1, 19–55. MR 0437541, DOI https://doi.org/10.7146/math.scand.a-11642
- Stacy G. Langton, Valuative criteria for families of vector bundles on algebraic varieties, Ann. of Math. (2) 101 (1975), 88–110. MR 0364255, DOI https://doi.org/10.2307/1970987
- Adrian Langer, Semistable sheaves in positive characteristic, Ann. of Math. (2) 159 (2004), no. 1, 251–276. MR 2051393, DOI https://doi.org/10.4007/annals.2004.159.251
- Shizhang Li, Semistable reduction for multi-filtered vector spaces, arXiv e-prints, 2017.
- Werner Lütkebohmert, Formal-algebraic and rigid-analytic geometry, Math. Ann. 286 (1990), no. 1-3, 341–371. MR 1032938, DOI https://doi.org/10.1007/BF01453580
- David Mumford, Abelian varieties, with appendices by C. P. Ramanujam and Yuri Manin, corrected reprint of the second (1974) edition, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay; by Hindustan Book Agency, New Delhi, 2008. MR 2514037
- Nitin Nitsure, Construction of Hilbert and Quot schemes, Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Amer. Math. Soc., Providence, RI, 2005, pp. 105–137. MR 2223407
- The Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu, 2018.
- M. Temkin, On local properties of non-Archimedean analytic spaces, Math. Ann. 318 (2000), no. 3, 585–607. MR 1800770, DOI https://doi.org/10.1007/s002080000123
- Harm Voskuil, Non-Archimedean Hopf surfaces, Sém. Théor. Nombres Bordeaux (2) 3 (1991), no. 2, 405–466. MR 1149806
- Evan Warner, Adic moduli spaces, PhD thesis, Stanford, https://searchworks.stanford.edu/view/12135003, 2017.
Additional Information
Shizhang Li
Affiliation:
Department of Mathematics, Columbia University, New York, New York 10027
Email:
shizhang@umich.edu
Received by editor(s):
November 8, 2018
Received by editor(s) in revised form:
January 11, 2019
Published electronically:
November 4, 2019
Article copyright:
© Copyright 2019
University Press, Inc.
