Abstract
Let \((X, \xi )\) be a polarized affine variety, i.e. an affine variety X with a (possibly irrational) Reeb vector field \(\xi \). We define the volume of a filtration of the coordinate ring of X in terms of the asymptotics of the average of jumping numbers. When the filtration is finitely generated, it induces a Fubini-Study function \(\varphi \) on the Berkovich analytification of X. In this case, we define the Monge–Ampère energy for \(\varphi \) using the theory of forms and currents on Berkovich spaces developed by Chambert-Loir and Ducros, and show that it agrees with the volume of the filtration. In the special case when the filtration comes from a test configuration, we recover the functional defined by Collins–Székelyhidi and Li–Xu.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berkovich, V.G.: Spectral Theory and Analytic Geometry Over Non-Archimedean Fields, vol. 33. American Mathematical Society, New York (2012)
Berman, R.J.: Conical Calabi-Yau metrics on toric affine varieties and convex cones. arXiv:2005.07053 (2020)
Berman, R., Boucksom, S., Jonsson, M.: A variational approach to the Yau–Tian–Donaldson conjecture. J. Am. Math. Soc. (2021)
Blum, H., Jonsson, M.: Thresholds, valuations, and K-stability. Adv. Math. 365, 107062 (2020)
Boucksom, S.: Corps d’Okounkov. Sémin. Bourbaki 65, 1–38 (2012)
Boucksom, S., Jonsson, M.: A non-Archimedean approach to K-stability. arXiv:1805.11160 (2018)
Boucksom, S., Jonsson, M.: Global pluripotential theory over a trivially valued field. arXiv:1801.08229v3 (2021)
Boucksom, Sébastien., Chen, Huayi: Okounkov bodies of filtered linear series. Compos. Math. 147(4), 1205–1229 (2011)
Boucksom, S., Favre, C., Jonsson, M.: A refinement of Izumi’s theorem. In: Valuation Theory in Interaction. EMS Series Congress Report. European Mathematical Society, Zürich (2014)
Boucksom, S., Hisamoto, T., Jonsson, M.: Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs. Ann. l’Institut Fourier 67(2), 743–841 (2017)
Chambert-Loir, A., Ducros, A.: Formes différentielles réelles et courants sur les espaces de Berkovich. arXiv:1204.6277 (2012)
Chen, X., Donaldson, S., Sun, S.: Kähler-Einstein metrics on Fano manifolds. J. Am. Math. Soc. 28(1), 183–197, 199–234, 235–278 (2015)
Collins, T.C., Székelyhidi, G.: K-semistability for irregular Sasakian manifolds. J. Differ. Geom. 109(1), 81–109 (2018)
Collins, T., Székelyhidi, G.: Sasaki-Einstein metrics and K-stability. Geom. Topol. 23(3), 1339–1413 (2019)
Cutkosky, S.: Multiplicities associated to graded families of ideals. Algebra Number Theory 7(9), 2059–2083 (2013)
Demailly, J.-P.: Complex Analytic and Differential Geometry. Citeseer, New York (1997)
Ein, L., Lazarsfeld, R., Smith, K.E.: Uniform approximation of Abhyankar valuation ideals in smooth function fields. Am. J. Math. 125(2), 409–440 (2003)
Fulton, W.: Introduction to Toric Varieties. Princeton University Press, Princeton (1993)
Gil, J.I.B., Gubler, W., Jell, P., Künnemann, K.: Pluripotential theory for tropical toric varieties and non-Archimedean Monge-Ampère equations. arXiv:2102.07392 (2021)
Kollár, J.: Seifert Gm-bundles. arXiv:math/0404386
Lagerberg, A.: Super currents and tropical geometry. Math. Z. 270(3–4), 1011–1050 (2012)
Lazarsfeld, R., Mustaţă, M.: Convex bodies associated to linear series. Ann. Sci. l’École Normale Supérieure 42, 783–835 (2009)
Li, C.: G-uniform stability and Kähler–Einstein metrics on Fano varieties. Invent. Math. 1–84. (2021)
Li, C., Chenyang, X.: Stability of valuations: higher rational rank. Peking Math. J. 1(1), 1–79 (2018)
Li, C., Wang, X., Xu, C.: Algebraicity of the metric tangent cones and equivariant K-stability. J. Am. Math. Soc. (2021)
Liu, Y., Xu, C., Zhuang, Z.: Finite generation for valuations computing stability thresholds and applications to K-stability. arXiv:2102.09405 (2021)
Martelli, D., Sparks, J., Yau, S.-T.: Sasaki-Einstein manifolds and volume minimisation. Commun. Math. Phys. 280(3), 611–673 (2008)
Nyström, D.W.: Test configurations and Okounkov bodies. Compos. Math. 148(6), 1736–1756 (2012)
Poineau, J.: Les espaces de Berkovich sont angéliques. Bull. Soc. Math. Fr. 141(2), 267–297 (2013)
Ross, J., Thomas, R.: Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics. J. Differ. Geom. 88(1), 109–159 (2011)
Tian, G.: K-stability and Kähler-Einstein metrics. Commun. Pure Appl. Math. 68(7), 1085–1156 (2015)
Vilsmeier, C.: A comparison of the real and non-archimedean Monge-Ampère operator. Math. Z. 297(1), 633–668 (2021)
Xu, C., Zhuang, Z.: Uniqueness of the minimizer of the normalized volume function. arXiv:2005.08303 (2020)
Acknowledgements
I would like to thank my advisor, Mattias Jonsson, for suggesting the problem and kindly sharing his ideas. I’m grateful to Chi Li for answering my questions on his papers. I thank Harold Blum, Sébastien Boucksom, Gabor Székelyhidi, Chenyang Xu and Ziquan Zhuang for helpful comments on a preliminary version of the paper. I’m also grateful to the anonymous referee for very helpful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Wu, Y. Volume and Monge–Ampère energy on polarized affine varieties. Math. Z. 301, 781–809 (2022). https://doi.org/10.1007/s00209-021-02935-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-021-02935-z