On the geometry of spaces of filtrations on local rings

References

[1] K. Ascher, D. Bejleri, H. Blum, K. DeVleming, D. Inchiostro, Y. Liu and X. Wang, Moduli of boundary polarized Calabi–Yau pairs, preprint (2023), https://arxiv.org/abs/2307.06522. Search in Google Scholar

[2] V. G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Math. Surveys Monogr. 33, American Mathematical Society, Providence 1990. Search in Google Scholar

[3] R. J. Berman, S. Boucksom, P. Eyssidieux, V. Guedj and A. Zeriahi, Kähler–Einstein metrics and the Kähler–Ricci flow on log Fano varieties, J. reine angew. Math. 751 (2019), 27–89. 10.1515/crelle-2016-0033Search in Google Scholar

[4] C. Birkar, P. Cascini, C. D. Hacon and J. McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405–468. 10.1090/S0894-0347-09-00649-3Search in Google Scholar

[5] H. Blum, Existence of valuations with smallest normalized volume, Compos. Math. 154 (2018), no. 4, 820–849. 10.1112/S0010437X17008016Search in Google Scholar

[6] H. Blum, Y. Liu and L. Qi, Convexity of multiplicities of filtrations on local rings, Compos. Math. 160 (2024), no. 4, 878–914. 10.1112/S0010437X23007972Search in Google Scholar

[7] H. Blum, Y. Liu, C. Xu and Z. Zhuang, The existence of the Kähler–Ricci soliton degeneration, Forum Math. Pi 11 (2023), Paper No. e9. 10.1017/fmp.2023.5Search in Google Scholar

[8] S. Bosch, U. Güntzer and R. Remmert, Non-Archimedean analysis, Grundlehren Math. Wiss. 261, Springer, Berlin 1984. 10.1007/978-3-642-52229-1Search in Google Scholar

[9] S. Boucksom, T. de Fernex and C. Favre, The volume of an isolated singularity, Duke Math. J. 161 (2012), no. 8, 1455–1520. 10.1215/00127094-1593317Search in Google Scholar

[10] S. Boucksom, T. de Fernex, C. Favre and S. Urbinati, Valuation spaces and multiplier ideals on singular varieties, Recent advances in algebraic geometry, London Math. Soc. Lecture Note Ser. 417, Cambridge University, Cambridge (2015), 29–51. 10.1017/CBO9781107416000.004Search in Google Scholar

[11] S. Boucksom, C. Favre and M. Jonsson, Valuations and plurisubharmonic singularities, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 449–494. 10.2977/prims/1210167334Search in Google Scholar

[12] S. Boucksom, C. Favre and M. Jonsson, A refinement of Izumi’s theorem, Valuation theory in interaction, EMS Ser. Congr. Rep., European Mathematical Society, Zürich (2014), 55–81. 10.4171/149-1/4Search in Google Scholar

[13] S. Boucksom, T. Hisamoto and M. Jonsson, Uniform K-stability, Duistermaat–Heckman measures and singularities of pairs, Ann. Inst. Fourier (Grenoble) 67 (2017), no. 2, 743–841. 10.5802/aif.3096Search in Google Scholar

[14] S. Boucksom and M. Jonsson, Global pluripotential theory over a trivially valued field, Ann. Fac. Sci. Toulouse Math. (6) 31 (2022), no. 3, 647–836. 10.5802/afst.1705Search in Google Scholar

[15] S. Boucksom and M. Jonsson, A non-Archimedean approach to K-stability, I: Metric geometry of spaces of test configurations and valuations, Ann. Inst. Fourier (Grenoble) 75 (2025), no. 2, 829–927. 10.5802/aif.3668Search in Google Scholar

[16] S. D. Cutkosky, Multiplicities associated to graded families of ideals, Algebra Number Theory 7 (2013), no. 9, 2059–2083. 10.2140/ant.2013.7.2059Search in Google Scholar

[17] S. D. Cutkosky, Asymptotic multiplicities of graded families of ideals and linear series, Adv. Math. 264 (2014), 55–113. 10.1016/j.aim.2014.07.004Search in Google Scholar

[18] S. D. Cutkosky and S. Praharaj, The asymptotic Samuel function of a filtration, Acta Math. Vietnam. 49 (2024), no. 1, 61–81. 10.1007/s40306-024-00523-xSearch in Google Scholar

[19] S. D. Cutkosky and P. Sarkar, Multiplicities and mixed multiplicities of arbitrary filtrations, Res. Math. Sci. 9 (2022), no. 1, Paper No. 14. 10.1007/s40687-021-00307-xSearch in Google Scholar

[20] T. Darvas, The Mabuchi geometry of finite energy classes, Adv. Math. 285 (2015), 182–219. 10.1016/j.aim.2015.08.005Search in Google Scholar

[21] T. Darvas, The Mabuchi completion of the space of Kähler potentials, Amer. J. Math. 139 (2017), no. 5, 1275–1313. 10.1353/ajm.2017.0032Search in Google Scholar

[22] T. Darvas, Geometric pluripotential theory on Kähler manifolds, Advances in complex geometry, Contemp. Math. 735, American Mathematical Society, Providence (2019), 1–104. 10.1090/conm/735/14822Search in Google Scholar

[23] T. Darvas and C. H. Lu, Geodesic stability, the space of rays and uniform convexity in Mabuchi geometry, Geom. Topol. 24 (2020), no. 4, 1907–1967. 10.2140/gt.2020.24.1907Search in Google Scholar

[24] T. Darvas and Y. A. Rubinstein, Tian’s properness conjectures and Finsler geometry of the space of Kähler metrics, J. Amer. Math. Soc. 30 (2017), no. 2, 347–387. 10.1090/jams/873Search in Google Scholar

[25] J.-P. Demailly and J. Kollár, Semi-continuity of complex singularity exponents and Kähler–Einstein metrics on Fano orbifolds, Ann. Sci. Éc. Norm. Supér. (4) 34 (2001), no. 4, 525–556. 10.1016/S0012-9593(01)01069-2Search in Google Scholar

[26] L. Ein, R. Lazarsfeld and K. E. Smith, Uniform approximation of Abhyankar valuation ideals in smooth function fields, Amer. J. Math. 125 (2003), no. 2, 409–440. 10.1353/ajm.2003.0010Search in Google Scholar

[27] C. Favre and M. Jonsson, The valuative tree, Lecture Notes in Math. 1853, Springer, Berlin 2004. 10.1007/b100262Search in Google Scholar

[28] S. Finski, Geometry at infinity of the space of Kähler potentials and asymptotic properties of filtrations, J. reine angew. Math. 818 (2025), 115–164. 10.1515/crelle-2024-0076Search in Google Scholar

[29] S. Finski, Submultiplicative norms and filtrations on section rings, Proc. Lond. Math. Soc. (3) 131 (2025), no. 2, Paper No. e70077. 10.1112/plms.70077Search in Google Scholar

[30] W. Fulton, Introduction to toric varieties, Ann. of Math. Stud. 131, Princeton University, Princeton 1993. 10.1515/9781400882526Search in Google Scholar

[31] C. Huneke and I. Swanson, Integral closure of ideals, rings, and modules, London Math. Soc. Lecture Note Ser. 336, Cambridge University, Cambridge 2006. Search in Google Scholar

[32] M. Jonsson and M. Mustaţă, Valuations and asymptotic invariants for sequences of ideals, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 6, 2145–2209. 10.5802/aif.2746Search in Google Scholar

[33] K. Kaveh and A. Khovanskii, Convex bodies and multiplicities of ideals, Proc. Steklov Inst. Math. 286 (2014), no. 1, 268–284. 10.1134/S0081543814060169Search in Google Scholar

[34] J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Math. 134, Cambridge University, Cambridge 1998. 10.1017/CBO9780511662560Search in Google Scholar

[35] R. Lazarsfeld and M. Mustaţă, Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 5, 783–835. 10.24033/asens.2109Search in Google Scholar

[36] C. Li, Minimizing normalized volumes of valuations, Math. Z. 289 (2018), no. 1–2, 491–513. 10.1007/s00209-017-1963-3Search in Google Scholar

[37] C. Li, Y. Liu and C. Xu, A guided tour to normalized volume, Geometric analysis—in honor of Gang Tian’s 60th birthday, Progr. Math. 333, Birkhäuser/Springer, Cham (2020), 167–219. 10.1007/978-3-030-34953-0_10Search in Google Scholar

[38] C. Li and C. Xu, Stability of valuations and Kollár components, J. Eur. Math. Soc. (JEMS) 22 (2020), no. 8, 2573–2627. 10.4171/jems/972Search in Google Scholar

[39] Y. Liu, The volume of singular Kähler–Einstein Fano varieties, Compos. Math. 154 (2018), no. 6, 1131–1158. 10.1112/S0010437X18007042Search in Google Scholar

[40] Y. Liu and C. Xu, A note on Kollár valuations. Special issue in memory of Gang Xiao, Sci. China Math. (2024), 1–8. 10.1007/s11425-024-2310-2Search in Google Scholar

[41] Y. Liu, C. Xu and Z. Zhuang, Finite generation for valuations computing stability thresholds and applications to K-stability, Ann. of Math. (2) 196 (2022), no. 2, 507–566. 10.4007/annals.2022.196.2.2Search in Google Scholar

[42] T. Mabuchi, Some symplectic geometry on compact Kähler manifolds. I, Osaka J. Math. 24 (1987), no. 2, 227–252. Search in Google Scholar

[43] M. Mustaţǎ, On multiplicities of graded sequences of ideals, J. Algebra 256 (2002), no. 1, 229–249. 10.1016/S0021-8693(02)00112-6Search in Google Scholar

[44] J. Peng, G m -equivariant degenerations of del Pezzo surfaces, preprint (2025), https://arxiv.org/abs/2503.06612. Search in Google Scholar

[45] R. Reboulet, Plurisubharmonic geodesics in spaces of non-Archimedean metrics of finite energy, J. reine angew. Math. 793 (2022), 59–103. 10.1515/crelle-2022-0059Search in Google Scholar

[46] R. Reboulet, The space of finite-energy metrics over a degeneration of complex manifolds, J. Éc. polytech. Math. 10 (2023), 659–701. 10.5802/jep.229Search in Google Scholar

[47] D. Rees, Valuations associated with ideals, Proc. Lond. Math. Soc. (3) 6 (1956), 161–174. 10.1112/plms/s3-6.2.161Search in Google Scholar

[48] D. Rees, Valuations associated with ideals. II, J. London Math. Soc. 31 (1956), 221–228. 10.1112/jlms/s1-31.2.221Search in Google Scholar

[49] Y. Wu, Volume and Monge–Ampère energy on polarized affine varieties, Math. Z. 301 (2022), no. 1, 781–809. 10.1007/s00209-021-02935-zSearch in Google Scholar

[50] M. Xia, Mabuchi geometry of big cohomology classes, J. reine angew. Math. 798 (2023), 261–292. 10.1515/crelle-2023-0019Search in Google Scholar

[51] C. Xu, A minimizing valuation is quasi-monomial, Ann. of Math. (2) 191 (2020), no. 3, 1003–1030. 10.4007/annals.2020.191.3.6Search in Google Scholar

[52] C. Xu, K-stability of Fano varieties, New Math. Monogr. 50, Cambridge University, Cambridge 2025. 10.1017/9781009538763Search in Google Scholar

[53] C. Xu and Z. Zhuang, Uniqueness of the minimizer of the normalized volume function, Camb. J. Math. 9 (2021), no. 1, 149–176. 10.4310/CJM.2021.v9.n1.a2Search in Google Scholar

[54] C. Xu and Z. Zhuang, Stable degenerations of singularities, J. Amer. Math. Soc. 38 (2025), no. 3, 585–626. 10.1090/jams/1055Search in Google Scholar

[55] Z. Zhuang, On boundedness of singularities and minimal log discrepancies of Kollár components, J. Algebraic Geom. 33 (2024), no. 3, 521–565. 10.1090/jag/822Search in Google Scholar

[56] Z. Zhuang, Stability of klt singularities, Handbook of geometry and topology of singularities VII, Springer, Cham (2025), 501–551. 10.1007/978-3-031-68711-2_10Search in Google Scholar