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Expectations, concave transforms, chow weights and Roth’s theorem for varieties

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Abstract

We present a new perspective, which is at the intersection of K-stability and Diophantine arithmetic geometry. It builds on work of Boucksom and Chen (Compos Math 147(4):1205–1229, 2011), Boucksom et al. (Math Ann 361(3–4):811–834, 2015), Boucksom et al. (Ann Inst Fourier (Grenoble) 67(2):743–841, 2017), Codogni and Patakfalvi (Invent Math 223(3):811–894, 2021), Fujita (Amer J Math 140(2):391–414, 2018), Fujita (J reine angew Math 751:309–338, 2019), Li (Duke Math J 166(16):3147–3218, 2017), Ferretti (Compos Math 121(3):247–262, 2000), Ferretti (Duke Math J 118(3): 493–522, 2003), McKinnon and Roth (Invent Math 200(2): 513–583, 2015), Ru and Vojta (Amer J Math 142(3):957–991, 2020), Ru and Wang (Algebra Number Theory 11(10): 2323–2337, 2017), Ru and Wang (Int J Number Theory 18(1):61–74, 2022), Heier and Levin (Amer J Math 143(1): 213–226, 2021) and our related works (including Grieve (Michigan Math J 67(2):371–404, 2018), Grieve (Houston J Math 44(4):1181–1202, 2018), Grieve (Asian J Math 24(6):995–1005, 2020), Grieve (Acta Arith 195(4):415–428, 2020), Grieve (Res Number Theory 7(1):1, 14, 2021) and Grieve (C R Math Acad Sci Soc R Can 44(1):16–32, 2022)). Our main results may be described via the expectations of the Duistermaat-Heckman measures. For example, we prove that the expected orders of vanishing for very ample linear series along subvarieties may be calculated via the theory of Chow weights and Okounkov bodies. This result provides a novel unified viewpoint to the existing works that surround these topics. It expands on work of Mumford (Enseignement Math (2) 23(1-2):39–110, 1977), Odaka (Ann of Math (2) 177(2):645–661, 2013) and Fujita (Amer J Math 140(2):391–414, 2018). Moreover, it has applications for Diophantine approximation and K-stability. As one result of this flavour, we define a concept of uniform arithmetic \(\mathrm {K}\)-stability for Kawamata log terminal pairs. We then establish a form of K. F.  Roth’s celebrated approximation theorem for certain collections of divisorial valuations which fail to be uniformly arithmetically \(\mathrm {K}\)-stable. This result builds on the Roth type theorems of McKinnon and Roth (Invent Math 200(2):513–583, 2015). Moreover, it complements the concept of arithmetic canonical boundedness, in the sense of McKinnon and Satriano (Trans Amer Math Soc 374(5):3557–3577, 2021), in addition to our results that establish instances of Vojta’s Main Conjecture for \({\mathbb {Q}}\)-Fano varieties Grieve (Asian J Math 24(6):995–1005, 2020).

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Acknowledgements

This work began while I was a postdoctoral fellow at the University of New Brunswick, Fredericton NB, where I was financially supported by an AARMS postdoctoral fellowship. It benefited from a visit to the Atlantic Algebra Centre, St. John’s NL, during March of 2017. Some later portions of this work were completed while I was a postdoctoral fellow at Michigan State University. They profited from visits to CIRGET, Montreal, during May of 2019, ICERM, Providence, during June of 2019, NCTS and Institute of Mathematics Academia Sinica, Taipei, during June of 2019, and the American Institute of Mathematics, San Jose, during January of 2020. Finally, I thank the Natural Sciences and Engineering Research Council of Canada for their support through my grants DGECR-2021-00218 and RGPIN-2021-03821 and colleagues and anonymous referees for their interest and helpful thoughtful comments.

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  1. Department of Mathematics & Computer Science, Royal Military College of Canada, P.O. Box 17000, Station Forces, Kingston, ON, K7K 7B4, Canada

    Nathan Grieve

  2. School of Mathematics and Statistics, 4302 Herzberg Laboratories, Carleton University, 1125 Colonel By Drive, Ottawa, ON, K1S 5B6, Canada

    Nathan Grieve

  3. Département de Mathématiques, Université du Québec à Montréal, Local PK-5151, 201 Avenue du Président-Kennedy, Montréal, QC, H2X 3Y7, Canada

    Nathan Grieve

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I thank the Natural Sciences and Engineering Research Council of Canada for their support through my grants DGECR-2021-00218 and RGPIN-2021-03821.

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Grieve, N. Expectations, concave transforms, chow weights and Roth’s theorem for varieties. manuscripta math. 172, 291–330 (2023). https://doi.org/10.1007/s00229-022-01417-8

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