Download this article
 Download this article For screen
For printing
Recent Issues

Volume 30
Issue 3, 835–1201
Issue 2, 389–833
Issue 1, 1–388

Volume 29, 9 issues

Volume 28, 9 issues

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
 
Subscriptions
 
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
 
Author index
To appear
 
Other MSP journals
Partial Okounkov bodies and Duistermaat–Heckman measures of non-Archimedean metrics

Mingchen Xia

Geometry & Topology 29 (2025) 1283–1344
Bibliography
1 O Amini, H Iriarte, Geometry of higher rank valuations, preprint (2022) arXiv:2208.06237
2 V G Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, 33, Amer. Math. Soc. (1990) MR1070709
3 R J Berman, B Berndtsson, Real Monge–Ampère equations and Kähler–Ricci solitons on toric log Fano varieties, Ann. Fac. Sci. Toulouse Math. 22 (2013) 649 MR3137248
4 R Berman, S Boucksom, Growth of balls of holomorphic sections and energy at equilibrium, Invent. Math. 181 (2010) 337 MR2657428
5 R J Berman, S Boucksom, M Jonsson, A variational approach to the Yau–Tian–Donaldson conjecture, J. Amer. Math. Soc. 34 (2021) 605 MR4334189
6 R Berman, S Boucksom, D Witt Nyström, Fekete points and convergence towards equilibrium measures on complex manifolds, Acta Math. 207 (2011) 1 MR2863909
7 A M Botero, J I Burgos Gil, D Holmes, R de Jong, Chern–Weil and Hilbert–Samuel formulae for singular Hermitian line bundles, Doc. Math. 27 (2022) 2563 MR4574244
8 S Boucksom, Cônes positifs des variétés complexes compactes, PhD thesis, Université Joseph-Fourier-Grenoble I (2002)
9 S Boucksom, Higher-dimensional Zariski decompositions, preprint (2002) arXiv:math/0204336
10 S Boucksom, Singularities of plurisubharmonic functions and multiplier ideals, preprint (2017)
11 S Boucksom, H Chen, Okounkov bodies of filtered linear series, Compos. Math. 147 (2011) 1205 MR2822867
12 S Boucksom, J P Demailly, M Păun, T Peternell, The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, J. Algebraic Geom. 22 (2013) 201 MR3019449
13 S Boucksom, P Eyssidieux, V Guedj, A Zeriahi, Monge–Ampère equations in big cohomology classes, Acta Math. 205 (2010) 199 MR2746347
14 S Boucksom, C Favre, M Jonsson, Valuations and plurisubharmonic singularities, Publ. Res. Inst. Math. Sci. 44 (2008) 449 MR2426355
15 S Boucksom, C Favre, M Jonsson, Differentiability of volumes of divisors and a problem of Teissier, J. Algebraic Geom. 18 (2009) 279 MR2475816
16 S Boucksom, T Hisamoto, M Jonsson, Uniform K-stability, Duistermaat–Heckman measures and singularities of pairs, Ann. Inst. Fourier (Grenoble) 67 (2017) 743 MR3669511
17 S Boucksom, M Jonsson, Global pluripotential theory over a trivially valued field, Ann. Fac. Sci. Toulouse Math. 31 (2022) 647 MR4452253
18 J I Burgos Gil, P Philippon, M Sombra, Arithmetic geometry of toric varieties: metrics, measures and heights, 360, Soc. Math. France (2014) MR3222615
19 J Cao, Numerical dimension and a Kawamata–Viehweg–Nadel-type vanishing theorem on compact Kähler manifolds, Compos. Math. 150 (2014) 1869 MR3279260
20 X Chen, J Cheng, On the constant scalar curvature Kähler metrics, III : General automorphism group, preprint (2018) arXiv:1801.05907
21 S R Choi, Y Hyun, J Park, J Won, Asymptotic base loci via Okounkov bodies, Adv. Math. 323 (2018) 784 MR3725891
22 S R Choi, Y Hyun, J Park, J Won, Okounkov bodies associated to pseudoeffective divisors, J. Lond. Math. Soc. 97 (2018) 170 MR3789843
23 C Ciliberto, M Farnik, A Küronya, V Lozovanu, J Roé, C Shramov, Newton–Okounkov bodies sprouting on the valuative tree, Rend. Circ. Mat. Palermo 66 (2017) 161 MR3694973
24 T Darvas, Geometric pluripotential theory on Kähler manifolds, from: "Advances in complex geometry" (editors Y A Rubinstein, B Shiffman), Contemp. Math. 735, Amer. Math. Soc. (2019) 1 MR3996485
25 T Darvas, E Di Nezza, C H Lu, L1 metric geometry of big cohomology classes, Ann. Inst. Fourier (Grenoble) 68 (2018) 3053 MR3959105
26 T Darvas, E Di Nezza, C H Lu, Monotonicity of nonpluripolar products and complex Monge–Ampère equations with prescribed singularity, Anal. PDE 11 (2018) 2049 MR3812864
27 T Darvas, E Di Nezza, C H Lu, On the singularity type of full mass currents in big cohomology classes, Compos. Math. 154 (2018) 380 MR3738831
28 T Darvas, E Di Nezza, C H Lu, Log-concavity of volume and complex Monge–Ampère equations with prescribed singularity, Math. Ann. 379 (2021) 95 MR4211083
29 T Darvas, E Di Nezza, H C Lu, The metric geometry of singularity types, J. Reine Angew. Math. 771 (2021) 137 MR4234098
30 T Darvas, C H Lu, Geodesic stability, the space of rays and uniform convexity in Mabuchi geometry, Geom. Topol. 24 (2020) 1907 MR4173924
31 T Darvas, R Reboulet, D Witt Nyström, M Xia, K Zhang, Transcendental Okounkov bodies, preprint (2023) arXiv:2309.07584
32 T Darvas, M Xia, The closures of test configurations and algebraic singularity types, Adv. Math. 397 (2022) 108198 MR4366232
33 T Darvas, M Xia, The volume of pseudoeffective line bundles and partial equilibrium, Geom. Topol. 28 (2024) 1957 MR4777706
34 J P Demailly, Analytic methods in algebraic geometry, 1, International (2012) MR2978333
35 J P Demailly, On the cohomology of pseudoeffective line bundles, from: "Complex geometry and dynamics" (editors J E Fornæss, M Irgens, E F Wold), Abel Symp. 10, Springer (2015) 51 MR3587462
36 J P Demailly, T Peternell, M Schneider, Pseudo-effective line bundles on compact Kähler manifolds, Int. J. Math. 12 (2001) 689 MR1875649
37 Y Deng, Transcendental Morse inequality and generalized Okounkov bodies, Algebr. Geom. 4 (2017) 177 MR3620634
38 K Fujiwara, F Kato, Foundations of rigid geometry, I, Eur. Math. Soc. (2018) MR3752648
39 V Guedj, A Zeriahi, Degenerate complex Monge–Ampère equations, 26, Eur. Math. Soc. (2017)
40 E Inoue, Entropies in μ-framework of canonical metrics and K-stability, II : Non-Archimedean aspect : non-Archimedean μ-entropy and μK-semistability, preprint (2022) arXiv:2202.12168
41 S Y Jow, Okounkov bodies and restricted volumes along very general curves, Adv. Math. 223 (2010) 1356 MR2581374
42 K Kaveh, Note on cohomology rings of spherical varieties and volume polynomial, J. Lie Theory 21 (2011) 263 MR2828718
43 K Kaveh, A G Khovanskii, Newton–Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, Ann. of Math. 176 (2012) 925 MR2950767
44 A G Khovanskii, Newton polyhedron, Hilbert polynomial and sums of finite sets, Funktsional. Anal. i Prilozhen. 26 (1992) 57 MR1209944
45 R Lazarsfeld, M Mustaţă, Convex bodies associated to linear series, Ann. Sci. École Norm. Sup. 42 (2009) 783 MR2571958
46 X Ma, G Marinescu, Holomorphic Morse inequalities and Bergman kernels, 254, Birkhäuser (2007) MR2339952
47 A Okounkov, Brunn–Minkowski inequality for multiplicities, Invent. Math. 125 (1996) 405 MR1400312
48 A Okounkov, Why would multiplicities be log-concave?, from: "The orbit method in geometry and physics" (editors C Duval, L Guieu, V Ovsienko), Progr. Math. 213, Birkhäuser (2003) 329 MR1995384
49 J Ross, D Witt Nyström, Analytic test configurations and geodesic rays, J. Symplectic Geom. 12 (2014) 125 MR3194078
50 R Schneider, Convex bodies : the Brunn–Minkowski theory, 151, Cambridge Univ. Press (2014) MR3155183
51 D Witt Nyström, Test configurations and Okounkov bodies, Compos. Math. 148 (2012) 1736 MR2999302
52 D Witt Nyström, Transforming metrics on a line bundle to the Okounkov body, Ann. Sci. École Norm. Sup. 47 (2014) 1111 MR3297156
53 D Witt Nyström, Monotonicity of non-pluripolar Monge–Ampère masses, Indiana Univ. Math. J. 68 (2019) 579 MR3951074
54 M Xia, Mabuchi geometry of big cohomology classes, J. Reine Angew. Math. 798 (2023) 261 MR4579706
55 M Xia, Pluripotential-theoretic stability thresholds, Int. Math. Res. Not. 2023 (2023) 12324 MR4615232
56 M Xia, Nonpluripolar products on vector bundles and Chern–Weil formulae, Math. Ann. 390 (2024) 3239 MR4803453
57 V P Zaharjuta, Transfinite diameter, Chebyshev constants, and capacity for compacta in n, Mat. Sb. 96(138) (1975) 374 MR0486623