Quantization in geometric pluripotential theory. (English) Zbl 1445.53062
Given an ample line bundle \(L\) over a compact \(K\)-manifold \((X, \omega )\) a central problem of Kähler geometry is to approximate, or to quantize, the infinite-dimensional space of Kähler potentials \(\mathcal{H}_{\omega}:=\{u\in C^{\infty}(X); \omega _u:=\omega +\sqrt{-1}\partial \overline{\partial}u>0\}\) with finite-dimensional spaces \(\mathcal{H}_k\) of positive Hermitian forms on \(H^0(X, L^k)\) since the \(\mathcal{H}_k\) can be identified as subspaces of \(\mathcal{H}_{\omega}\) consisting of algebraic Fubini-Study metrics. This paper considers natural \(L^p\) Finsler geometries on \(\mathcal{H}_k\) and proves that the resulting complete path-lenght metric spaces \((\mathcal{H}_k, d_{p, k})\) approximate and recover \((\mathcal{E}^p_{\omega}, d_p)\). Here, \((\mathcal{E}^p_{\omega}, d_p)\) is the completition of path-length metric structures \((\mathcal{H}_{\omega}, d_p)\) recently obtained through some \(L^p\)-type Finsler structures on \(\mathcal{H}_{\omega}\). This main result has a number of interesting applications, such as a new Lidskii-type inequality on the space of Kähler metrics and a new approach to Pythagorean-Kähler formulas.
Reviewer: Mircea Crâşmăreanu (Iaşi)
MSC:
| 53D50 | Geometric quantization |
| 58D17 | Manifolds of metrics (especially Riemannian) |
| 58B20 | Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds |
References:
| [1] | Auvray, H.; Ma, X.; Marinescu, G.Bergman kernels on punctured Riemann surfaces. C. R. Math. Acad. Sci. Paris354 (2016), no. 10, 1018-1022. doi: 10.1016/j.crma.2016.08.006 · Zbl 1354.32002 |
| [2] | Bedford, E.; Taylor, B. A.A new capacity for plurisubharmonic functions. Acta Math.149 (1982), no. 1‐2, 1-40. doi: 10.1007/BF02392348 · Zbl 0547.32012 |
| [3] | Berman, R. J.Bergman kernels and equilibrium measures for line bundles over projective manifolds. Amer. J. Math.131 (2009), no. 5, 1485-1524. doi: 10.1353/ajm.0.0077 · Zbl 1191.32008 |
| [4] | Berman, R. J.From Monge-Ampère equations to envelopes and geodesic rays in the zero temperature limit. Math. Z. (2018). doi: 10.1007/s00209-018-2087-0 · Zbl 1437.32007 |
| [5] | Berman, R.; Berndtsson, B.; Sjöstrand, J.A direct approach to Bergman kernel asymptotics for positive line bundles. Ark. Mat.46 (2008), no. 2, 197-217. doi: 10.1007/s11512-008-0077-x · Zbl 1161.32001 |
| [6] | Berman, R.; Boucksom, S.; Jonsson, M. A variational approach to the Yau‐Tian‐Donaldson conjecture. Preprint, 2015a. arXiv:1509.04561 [math.DG] |
| [7] | Berman, R. J.; Darvas, T.; Lu, C. H. Regularity of weak minimizers of the K‐energy and applications to properness and K‐stability. Annales de l’ENS, forthcoming. Preprint, 2015b. arXiv:1509.04561 [math.DG] |
| [8] | Berman, R. J.; Darvas, T.; Lu, C. H.Convexity of the extended K‐energy and the large time behavior of the weak Calabi flow. Geom. Topol.21 (2017), no. 5, 2945-2988. doi: 10.2140/gt.2017.21.2945 · Zbl 1372.53073 |
| [9] | Berman, R. J.; Keller, J. Bergman geodesics. Bergman geodesics. Complex Monge‐Ampére equations and geodesics in the space of Kähler metrics, 283-302. Lecture Notes in Mathematics, 2038. Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-23669-3_8 · Zbl 1231.32002 |
| [10] | Berndtsson, B. Probability measures related to geodesics in the space of Kähler metrics. Preprint, 2009. arXiv:0907.1806 [math.DG] |
| [11] | Berndtsson, B.; Lempert, L.A proof of the Ohsawa‐Takegoshi theorem with sharp estimates. J. Math. Soc. Japan68 (2016), no. 4, 1461-1472. doi: 10.2969/jmsj/06841461 · Zbl 1360.32006 |
| [12] | Bhatia, R.Matrix analysis. Graduate Texts in Mathematics, 169. Springer, New York, 1997. doi: 10.1007/978-1-4612-0653-8 |
| [13] | Błocki, Z. On geodesics in the space of Kähler metrics. Advances in geometric analysis, 3-19. Advanced Lectures in Mathematics (ALM), 21. International Press, Somerville, Mass., 2012. · Zbl 1329.32009 |
| [14] | Błocki, Z.; Kołodziej, S.On regularization of plurisubharmonic functions on manifolds. Proc. Amer. Math. Soc.135 (2007), no. 7, 2089-2093. doi: 10.1090/S0002-9939-07-08858-2 · Zbl 1116.32024 |
| [15] | Bouche, T. Convergence de la métrique de Fubini‐Study d’un fibré linéaire positif. Ann. Inst. Fourier (Grenoble)40 (1990), no. 1, 117-130. · Zbl 0685.32015 |
| [16] | Boucksom, S. Variational and non‐Archimedean aspects of the Yau-Tian-Donaldson conjecture. Preprint, 2018. arXiv:1805.03289 [math.DG] · Zbl 1447.32030 |
| [17] | Boucksom, S.; Eriksson, D. Spaces of norms, determinant of cohomology and Fekete points in non‐Archimedean geometry. Preprint, 2018. arXiv:1805.01016 [math.AG] |
| [18] | Boucksom, S.; Jonsson, M. A non‐Archimedean approach to K‐stability. Preprint, 2018. arXiv:1805.11160 [math.AG] |
| [19] | Boutet de Monvel, L.; Sjöstrand, J. Sur la singularité des noyaux de Bergman et de Szegő. Journées: Équations aux dérivées partielles de rennes (1975), 123-164. Astérisque, 34-35. Soc. Math. France, Paris, 1976. · Zbl 0344.32010 |
| [20] | Carlen, E. Trace inequalities and quantum entropy: an introductory course. Entropy and the quantum, 73-140. Contemporary Mathematics, 529. American Mathematical Society, Providence, R.I., 2010. doi: 10.1090/conm/529/10428 · Zbl 1218.81023 |
| [21] | Catlin, D. The Bergman kernel and a theorem of Tian. Analysis and geometry in several complex variables (Katata, 1997), 1-23. Trends in Mathematics. Birkhäuser Boston, Boston, 1999. · Zbl 0941.32002 |
| [22] | Chen, X. The space of Kähler metrics. J. Differential Geom.56 (2000), no. 2, 189-234. · Zbl 1041.58003 |
| [23] | Chen, X.; Cheng, J. On the constant scalar curvature Kähler metrics, existence results. Preprint, 2018a. arXiv:1801.00656 [math.DG] |
| [24] | Chen, X.; Cheng, J. On the constant scalar curvature Kähler metrics, general automorphism group. Preprint, 2018b. arXiv:1801.05907 [math.DG] |
| [25] | Chen, X.; Sun, S. Space of Kähler metrics (V)—Kähler quantization. Metric and differential geometry, 19-41. Progress in Mathematics, 297. Birkhäuser/Springer, Basel, 2012. doi: 10.1007/978-3-0348-0257-4_2 · Zbl 1252.53084 |
| [26] | Coman, D.; Ma, X.; Marinescu, G.Equidistribution for sequences of line bundles on normal Kähler spaces. Geom. Topol.21 (2017), no. 2, 923-962. doi: 10.2140/gt.2017.21.923 · Zbl 1457.32051 |
| [27] | Coman, D.; Marinescu, G.Equidistribution results for singular metrics on line bundles. Ann. Sci. Éc. Norm. Supér. (4)48 (2015), no. 3, 497-536. doi: 10.24033/asens.2250 · Zbl 1364.32014 |
| [28] | Darvas, T. The Mabuchi geometry of finite energy classes. Adv. Math.285 (2015), 182-219. doi: 10.1016/j.aim.2015.08.005 · Zbl 1327.53093 |
| [29] | Darvas, T. The Mabuchi completion of the space of Kähler potentials. Amer. J. Math.139 (2017), no. 5, 1275-1313. doi: 10.1353/ajm.2017.0032 · Zbl 1387.53095 |
| [30] | Darvas, T. Geometric pluripotential theory on Kähler manifolds. Preprint, 2018. Available at: https://www.math.umd.edu/ tdarvas/items/Geometric_Pluripotential_Theory.pdf. |
| [31] | Darvas, T.; Rubinstein, Y. A.Kiselman’s principle, the Dirichlet problem for the Monge‐Ampère equation, and rooftop obstacle problems. J. Math. Soc. Japan68 (2016), no. 2, 773-796. doi: 10.2969/jmsj/06820773 · Zbl 1353.32039 |
| [32] | Darvas, T.; Rubinstein, Y. A.Tian’s properness conjectures and Finsler geometry of the space of Kähler metrics. J. Amer. Math. Soc.30 (2017), no. 2, 347-387. doi: 10.1090/jams/873 · Zbl 1386.32021 |
| [33] | Demailly, J.‐P.Regularization of closed positive currents and intersection theory. J. Algebraic Geom.1 (1992), no. 3, 361-409. · Zbl 0777.32016 |
| [34] | Demailly, J.‐P.Complex analytic and differential geometry. Course, 2018. Available at: https://www‐fourier.ujf‐grenoble.fr/ demailly/manuscripts/agbook.pdf. |
| [35] | Dervan, R.; Ross, J.K‐stability for Kähler manifolds. Math. Res. Lett.24 (2017), no. 3, 689-739. doi: 10.4310/MRL.2017.v24.n3.a5 · Zbl 1390.32021 |
| [36] | Dinew, S.Uniqueness in ℰ(X,ω). J. Funct. Anal.256 (2009), no. 7, 2113-2122. doi: 10.1016/j.jfa.2009.01.019 · Zbl 1171.32024 |
| [37] | Dinh, T.‐C.; Ma, X.; Marinescu, G.Equidistribution and convergence speed for zeros of holomorphic sections of singular Hermitian line bundles. J. Funct. Anal.271 (2016), no. 11, 3082-3110. doi: 10.1016/j.jfa.2016.09.007 · Zbl 1373.32016 |
| [38] | Donaldson, S. K. Symmetric spaces, Kähler geometry and Hamiltonian dynamics. Northern California Symplectic Geometry Seminar, 13-33. American Mathematical Society Translations, Series 2, 196. Adv. Math. Sci., 45. American Mathematical Society, Providence, R.I., 1999. doi: 10.1090/trans2/196/02 · Zbl 0972.53025 |
| [39] | Donaldson, S. K.Scalar curvature and projective embeddings. I. J. Differential Geom.59 (2001), no. 3, 479-522. Available at: http://projecteuclid.org/euclid.jdg/1090349449 · Zbl 1052.32017 |
| [40] | Donaldson, S. K.Scalar curvature and projective embeddings. II. Q. J. Math.56 (2005), no. 3, 345-356. doi: 10.1093/qmath/hah044 · Zbl 1159.32012 |
| [41] | Guedj, V. The metric completion of the Riemannian space of Kähler metrics. Preprint, 2014. arXiv:1401.7857 [math.DG] |
| [42] | Guedj, V.; Zeriahi, A.Intrinsic capacities on compact Kähler manifolds. J. Geom. Anal.15 (2005), no. 4, 607-639. doi: 10.1007/BF02922247 · Zbl 1087.32020 |
| [43] | Guedj, V.; Zeriahi, A.The weighted Monge‐Ampère energy of quasiplurisubharmonic functions. J. Funct. Anal.250 (2007), no. 2, 442-482. doi: 10.1016/j.jfa.2007.04.018 · Zbl 1143.32022 |
| [44] | Guedj, V.; Zeriahi, A.Degenerate complex Monge‐Ampère equations. EMS Tracts in Mathematics, 26. European Mathematical Society (EMS), Zürich, 2017. doi: 10.4171/167 · Zbl 1373.32001 |
| [45] | Kapovich, M.; Leeb, B.Finsler bordifications of symmetric and certain locally symmetric spaces. Geom. Topol.22 (2018), no. 5, 2533-2646. doi: 10.2140/gt.2018.22.2533 · Zbl 1417.53058 |
| [46] | Kempf, G. R. Metrics on invertible sheaves on abelian varieties. Topics in algebraic geometry (Guanajuato, 1989), 107-108. Aportaciones Mat. Notas Investigación, 5. Soc. Mat. Mexicana, México, 1992. · Zbl 0866.14028 |
| [47] | Kobayashi, S.Differential geometry of complex vector bundles. Princeton Legacy Library. Princeton University Press, Princeton, N.J., 2014. |
| [48] | Lu, Z. On the lower order terms of the asymptotic expansion of Tian‐Yau‐Zelditch. Amer. J. Math.122 (2000), no. 2, 235-273. · Zbl 0972.53042 |
| [49] | Ma, X.; Marinescu, G.Holomorphic Morse inequalities and Bergman kernels. Progress in Mathematics, 254. Birkhäuser, Basel, 2007. · Zbl 1135.32001 |
| [50] | Mabuchi, T.Some symplectic geometry on compact Kähler manifolds. I. Osaka J. Math.24 (1987), no. 2, 227-252. · Zbl 0645.53038 |
| [51] | Ohsawa, T.; Takegoshi, K.On the extension of L^2 holomorphic functions. Math. Z.195 (1987), no. 2, 197-204. doi: 10.1007/BF01166457 · Zbl 0625.32011 |
| [52] | Phong, D. H.; Song, J.; Sturm, J.; Weinkove, B.The Moser‐Trudinger inequality on Kähler‐Einstein manifolds. Amer. J. Math.130 (2008), no. 4, 1067-1085. doi: 10.1353/ajm.0.0013 · Zbl 1158.58005 |
| [53] | Phong, D. H.; Sturm, J.The Monge‐Ampère operator and geodesics in the space of Kähler potentials. Invent. Math.166 (2006), no. 1, 125-149. doi: 10.1007/s00222-006-0512-1 · Zbl 1120.32026 |
| [54] | Rao, M. M.; Ren, Z. D. Theory of Orlicz spaces. Monographs and Textbooks in Pure and Applied Mathematics, 146. Marcel Dekker, New York, 1991. · Zbl 0724.46032 |
| [55] | Ross, J.; Witt Nyström, D.Analytic test configurations and geodesic rays. J. Symplectic Geom.12 (2014), no. 1, 125-169. doi: 10.4310/JSG.2014.v12.n1.a5 · Zbl 1300.32021 |
| [56] | Ruan, W.‐D.Canonical coordinates and Bergmann [Bergman] metrics. Comm. Anal. Geom.6 (1998), no. 3, 589-631. doi: 10.4310/CAG.1998.v6.n3.a5 · Zbl 0917.53026 |
| [57] | Rubinstein, Y. A. Tian’s properness conjectures: an introduction to Kähler geometry. Preprint, 2018. arXiv:1807.00928 [math.DG] · Zbl 1444.53001 |
| [58] | Semmes, S. Interpolation of Banach spaces, differential geometry and differential equations. Rev. Mat. Iberoamericana4 (1988), no. 1, 155-176. doi: 10.4171/RMI/67 · Zbl 0716.46056 |
| [59] | Semmes, S. Complex Monge‐Ampère and symplectic manifolds. Amer. J. Math.114 (1992), no. 3, 495-550. doi: 10.2307/2374768 · Zbl 0790.32017 |
| [60] | Sjöström Dyrefelt, Z. On K‐Polystability of cscK manifolds with transcendental cohomology class. International Mathematics Research Notices (2018). doi: 10.1093/imrn/rny094 · Zbl 1409.32017 |
| [61] | Song, J.; Zelditch, S.Bergman metrics and geodesics in the space of Kähler metrics on toric varieties. Anal. PDE3 (2010), no. 3, 295-358. doi: 10.2140/apde.2010.3.295 · Zbl 1282.35428 |
| [62] | Tian, G.Kahler metrics on algebraic manifolds. Ph.D. thesis, Harvard University, 1988. |
| [63] | Tian, G. On a set of polarized Kähler metrics on algebraic manifolds. J. Differential Geom.32 (1990), no. 1, 99-130. · Zbl 0706.53036 |
| [64] | Tian, G.; Zhu, X.A nonlinear inequality of Moser‐Trudinger type. Calc. Var. Partial Differential Equations10 (2000), no. 4, 349-354. doi: 10.1007/s005260010349 · Zbl 0987.32017 |
| [65] | Yau, S.‐T. Nonlinear analysis in geometry. Monographies de L’Enseignement Mathématique, 33. Série des Conférences de l’Union Mathématique Internationale, 8. L’Enseignement Mathématique, Geneva, 1986. 54 pp. · Zbl 0631.53003 |
| [66] | Zelditch, S.Szegő kernels and a theorem of Tian. Internat. Math. Res. Notices (1998), no. 6, 317-331. doi: 10.1155/S107379289800021X · Zbl 0922.58082 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
