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

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).