Stability of measures on Kähler manifolds. (English) Zbl 1355.53072
Summary: Let \((M, \omega)\) be a Kähler manifold and let \(K\) be a compact group that acts on \(M\) in a Hamiltonian fashion. We study the action of \(K^{\mathbb{C}}\) on probability measures on \(M\). First of all we identify an abstract setting for the momentum mapping and give numerical criteria for stability, semi-stability and polystability. Next we apply this setting to the action of \(K^{\mathbb{C}}\) on measures. We get various stability criteria for measures on Kähler manifolds. The same circle of ideas gives a very general surjectivity result for a map originally studied by Hersch and Bourguignon-Li-Yau.
MSC:
| 53D20 | Momentum maps; symplectic reduction |
| 32M05 | Complex Lie groups, group actions on complex spaces |
| 14L24 | Geometric invariant theory |
References:
| [1] | Apostolov, V.; Jakobson, D.; Kokarev, G., An extremal eigenvalue problem in Kähler geometry, J. Geom. Phys., 91, 108-116 (2015) · Zbl 1325.49054 |
| [2] | Arezzo, C.; Ghigi, A.; Loi, A., Stable bundles and the first eigenvalue of the Laplacian, J. Geom. Anal., 17, 3, 375-386 (2007) · Zbl 1128.58013 |
| [3] | Atiyah, M. F., Convexity and commuting Hamiltonians, Bull. Lond. Math. Soc., 14, 1, 1-15 (1982) · Zbl 0482.58013 |
| [4] | Audin, M., Torus Actions on Symplectic Manifolds, Progr. Math., vol. 93 (2004), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 1062.57040 |
| [5] | Azad, H.; Loeb, J.-J., Plurisubharmonic functions and the Kempf-Ness theorem, Bull. Lond. Math. Soc., 25, 2, 162-168 (1993) · Zbl 0795.32002 |
| [6] | Berger, M., Geometry. I, Universitext (1987), Springer-Verlag: Springer-Verlag Berlin, translated from the French by M. Cole and S. Levy · Zbl 0606.51001 |
| [7] | Biliotti, L.; Ghigi, A., Homogeneous bundles and the first eigenvalue of symmetric spaces, Ann. Inst. Fourier (Grenoble), 58, 7, 2315-2331 (2008) · Zbl 1161.53064 |
| [8] | Biliotti, L.; Ghigi, A., Satake-Furstenberg compactifications, the moment map and \(\lambda_1\), Amer. J. Math., 135, 1, 237-274 (2013) · Zbl 1261.53050 |
| [9] | Biliotti, L.; Ghigi, A.; Heinzner, P., Invariant convex sets in polar representations, Israel J. Math., 213, 1, 423-441 (June 2016) · Zbl 1351.52002 |
| [10] | Borel, A.; Ji, L., Compactifications of Symmetric and Locally Symmetric Spaces, Math. Theory Appl. (2006), Birkhäuser Boston Inc.: Birkhäuser Boston Inc. Boston, MA · Zbl 1100.22001 |
| [11] | Bourguignon, J.-P.; Li, P.; Yau, S.-T., Upper bound for the first eigenvalue of algebraic submanifolds, Comment. Math. Helv., 69, 2, 199-207 (1994) · Zbl 0814.53040 |
| [12] | Donaldson, S. K., Some numerical results in complex differential geometry, Pure Appl. Math. Q., 5, 2, Special Issue: In honor of Friedrich Hirzebruch. Part 1, 571-618 (2009) · Zbl 1178.32018 |
| [13] | Dugundji, J., Topology (1966), Allyn and Bacon Inc.: Allyn and Bacon Inc. Boston, Mass · Zbl 0144.21501 |
| [14] | Duistermaat, J. J.; Kolk, J. A.C., Lie Groups, Universitext (2000), Springer-Verlag: Springer-Verlag Berlin · Zbl 0955.22001 |
| [15] | Dunford, N.; Schwartz, J. T., Linear Operators. I. General Theory, Pure Appl. Math., vol. 7 (1958), Interscience Publishers, Inc./Interscience Publishers, Ltd.: Interscience Publishers, Inc./Interscience Publishers, Ltd. New York/London, with the assistance of W.G. Bade and R.G. Bartle · Zbl 0084.10402 |
| [16] | Folland, G. B., Real Analysis, Pure Appl. Math. (1999), John Wiley & Sons: John Wiley & Sons New York · Zbl 0671.58036 |
| [17] | Gangbo, W.; Kim, H. K.; Pacini, T., Differential forms on Wasserstein space and infinite-dimensional Hamiltonian systems, Mem. Amer. Math. Soc., 211, 993 (2011), vi+77 · Zbl 1221.53001 |
| [18] | Georgulas, V.; Robbin, J. W.; Salamon, D. A., The moment-weight inequality and the Hilbert-Mumford criterion (2013), preprint |
| [19] | Guillemin, V., Moment Maps and Combinatorial Invariants of Hamiltonian \(T^n\)-Spaces, Progr. Math., vol. 122 (1994), Birkhäuser Boston Inc.: Birkhäuser Boston Inc. Boston, MA · Zbl 0828.58001 |
| [20] | Guillemin, V.; Sternberg, S., Convexity properties of the moment mapping, Invent. Math., 67, 3, 491-513 (1982) · Zbl 0503.58017 |
| [21] | Heinzner, P., Geometric invariant theory on Stein spaces, Math. Ann., 289, 4, 631-662 (1991) · Zbl 0728.32010 |
| [22] | Heinzner, P.; Huckleberry, A., Kählerian potentials and convexity properties of the moment map, Invent. Math., 126, 1, 65-84 (1996) · Zbl 0855.58025 |
| [23] | Heinzner, P.; Huckleberry, A., Analytic Hilbert quotients, (Several Complex Variables. Several Complex Variables, Berkeley, CA, 1995-1996. Several Complex Variables. Several Complex Variables, Berkeley, CA, 1995-1996, Math. Sci. Res. Inst. Publ., vol. 37 (1999), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 309-349 · Zbl 0959.32013 |
| [24] | Heinzner, P.; Huckleberry, A. T.; Loose, F., Kählerian extensions of the symplectic reduction, J. Reine Angew. Math., 455, 123-140 (1994) · Zbl 0803.53042 |
| [25] | Heinzner, P.; Loose, F., Reduction of complex Hamiltonian \(G\)-spaces, Geom. Funct. Anal., 4, 3, 288-297 (1994) · Zbl 0816.53018 |
| [26] | Heinzner, P.; Schwarz, G. W., Cartan decomposition of the moment map, Math. Ann., 337, 1, 197-232 (2007) · Zbl 1110.32008 |
| [27] | Heinzner, P.; Schwarz, G. W.; Stötzel, H., Stratifications with respect to actions of real reductive groups, Compos. Math., 144, 1, 163-185 (2008) · Zbl 1133.32009 |
| [28] | Heinzner, P.; Stötzel, H., Critical points of the square of the momentum map, (Global Aspects of Complex Geometry (2006), Springer: Springer Berlin), 211-226 · Zbl 1117.32017 |
| [29] | Hersch, J., Quatre propriétés isopérimétriques de membranes sphériques homogènes, C. R. Acad. Sci. Paris Sér. A-B, 270, A1645-A1648 (1970) · Zbl 0224.73083 |
| [30] | Hirsch, M. W., Differential Topology, Grad. Texts in Math., vol. 33 (1976), Springer-Verlag: Springer-Verlag New York · Zbl 0121.18004 |
| [31] | Kapovich, M.; Leeb, B.; Millson, J., Convex functions on symmetric spaces, side lengths of polygons and the stability inequalities for weighted configurations at infinity, J. Differential Geom., 81, 2, 297-354 (2009) · Zbl 1167.53044 |
| [32] | Kempf, G.; Ness, L., The length of vectors in representation spaces, (Algebraic Geometry, Proc. Summer Meeting. Algebraic Geometry, Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978. Algebraic Geometry, Proc. Summer Meeting. Algebraic Geometry, Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978, Lecture Notes in Math., vol. 732 (1979), Springer: Springer Berlin), 233-243 · Zbl 0407.22012 |
| [33] | Korányi, A., Remarks on the Satake compactifications, Pure Appl. Math. Q., 1, 4, part 3, 851-866 (2005) · Zbl 1158.22009 |
| [34] | Legendre, E.; Sena-Dias, R., Toric aspects of the first eigenvalues (2015), preprint |
| [35] | McDuff, D.; Salamon, D., Introduction to Symplectic Topology, Oxford Math. Monogr. (1998), The Clarendon Press Oxford University Press: The Clarendon Press Oxford University Press New York · Zbl 1066.53137 |
| [36] | Millson, J. J.; Zombro, B., A Kähler structure on the moduli space of isometric maps of a circle into Euclidean space, Invent. Math., 123, 1, 35-59 (1996) · Zbl 0859.58007 |
| [37] | Mumford, D.; Fogarty, J.; Kirwan, F., Geometric Invariant Theory, Ergeb. Math. Grenzgeb. (2), vol. 34 (1994), Springer-Verlag: Springer-Verlag Berlin · Zbl 0797.14004 |
| [38] | Mundet i. Riera, I., A Hitchin-Kobayashi correspondence for Kähler fibrations, J. Reine Angew. Math., 528, 41-80 (2000) · Zbl 1002.53057 |
| [39] | Mundet i. Riera, I., A Hilbert-Mumford criterion for polystability in Kaehler geometry, Trans. Amer. Math. Soc., 362, 10, 5169-5187 (2010) · Zbl 1201.53086 |
| [40] | Panelli, F.; Podestà, F., On the first eigenvalue of invariant Kähler metrics, Math. Z., 281, 1-2, 471-482 (2015) · Zbl 1325.53050 |
| [41] | Schneider, R., Convex Bodies: the Brunn-Minkowski Theory, Encyclopedia Math. Appl., vol. 44 (1993), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0798.52001 |
| [42] | Sjamaar, R., Holomorphic slices, symplectic reduction and multiplicities of representations, Ann. of Math. (2), 141, 1, 87-129 (1995) · Zbl 0827.32030 |
| [43] | Teleman, A., Symplectic stability, analytic stability in non-algebraic complex geometry, Internat. J. Math., 15, 2, 183-209 (2004) · Zbl 1089.53058 |
| [44] | Thomas, R. P., Notes on GIT and symplectic reduction for bundles and varieties, (Surveys in Differential Geometry. Vol. X. Surveys in Differential Geometry. Vol. X, Surv. Differ. Geom., vol. 10 (2006), Int. Press: Int. Press Somerville, MA), 221-273 · Zbl 1132.14043 |
| [45] | Tian, G., Canonical Metrics in Kähler Geometry (2000), Birkhäuser Verlag: Birkhäuser Verlag Basel, notes taken by Meike Akveld · Zbl 0978.53002 |
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