On the complex structure of symplectic quotients. (English) Zbl 1489.53101
Summary: Let \(K\) be a compact group. For a symplectic quotient \(M_{\lambda}\) of a compact Hamiltonian Kähler \(K\)-manifold, we show that the induced complex structure on \(M_{\lambda}\) is locally invariant when the parameter \(\lambda\) varies in \(\mathrm{Lie}(K)^{\ast}\). To prove such a result, we take two different approaches: (i) use the complex geometry properties of the symplectic implosion construction; (ii) investigate the variation of geometric invariant theory (GIT) quotients.
MSC:
| 53C55 | Global differential geometry of Hermitian and Kählerian manifolds |
| 53D20 | Momentum maps; symplectic reduction |
| 14L24 | Geometric invariant theory |
References:
| [1] | Atiyah, M. F., Convexity and commuting Hamiltonians, Bull Lond Math Soc, 14, 1-15 (1982) · Zbl 0482.58013 · doi:10.1112/blms/14.1.1 |
| [2] | Białynicki-Birula, A., Finiteness of the number of maximal open subsets with good quotients, Transform Groups, 3, 301-319 (1998) · Zbl 0940.14034 · doi:10.1007/BF01234530 |
| [3] | Chen, X.; Sun, S., Calabi flow, geodesic rays, and uniqueness of constant scalar curvature Kähler metrics, Ann of Math (2), 180, 407-454 (2014) · Zbl 1307.53058 · doi:10.4007/annals.2014.180.2.1 |
| [4] | Dolgachev, I.; Hu, Y., Variation of geometric invariant theory quotients, Publ Math Inst Hautes Études Sci, 87, 5-51 (1998) · Zbl 1001.14018 · doi:10.1007/BF02698859 |
| [5] | Futaki, A., Kähler-Einstein Metrics and Integral Invariants (1988), Berlin: Springer-Verlag, Berlin · Zbl 0646.53045 · doi:10.1007/BFb0078084 |
| [6] | Georgoulas V, Robbin J W, Salamon D A. The moment-weight inequality and the Hilbert-Mumford criterion. arX-iv:1311.0410, 2013 · Zbl 1492.14002 |
| [7] | Greb, D.; Heinzner, P., Kählerian reduction in steps, Symmetry and Spaces, 63-82 (2010), Boston: Birkhäuser, Boston · Zbl 1203.53083 · doi:10.1007/978-0-8176-4875-6_5 |
| [8] | Guillemin, V.; Jeffrey, L.; Sjamaar, R., Symplectic implosion, Transform Groups, 7, 155-184 (2002) · Zbl 1015.53054 · doi:10.1007/s00031-002-0009-y |
| [9] | Guillemin, V.; Sternberg, S., Geometric quantization and multiplicities of group representations, Invent Math, 67, 515-538 (1982) · Zbl 0503.58018 · doi:10.1007/BF01398934 |
| [10] | Heinzner, P.; Huckleberry, A., Kählerian potentials and convexity properties of the moment map, Invent Math, 126, 65-84 (1996) · Zbl 0855.58025 · doi:10.1007/s002220050089 |
| [11] | Heinzner, P.; Huckleberry, A., Analytic Hilbert quotients, Several Complex Variables, 309-349 (1999), Cambridge: Cambridge University Press, Cambridge · Zbl 0959.32013 |
| [12] | Heinzner, P.; Loose, F., Reduction of complex Hamiltonian G-spaces, Geom Funct Anal, 4, 288-297 (1994) · Zbl 0816.53018 · doi:10.1007/BF01896243 |
| [13] | Heinzner, P.; Migliorini, L., Projectivity of moment map quotients, Osaka J Math, 38, 167-184 (2001) · Zbl 0982.32020 |
| [14] | Hu, Y., (W, R)-matroids and thin Schubert-type cells attached to algebraic torus actions, Proc Amer Math Soc, 123, 2607-2617 (1995) · Zbl 0867.14020 |
| [15] | 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, 297-354 (2009) · Zbl 1167.53044 · doi:10.4310/jdg/1231856263 |
| [16] | Kirwan, F., Cohomology of Quotients in Symplectic and Algebraic Geometry (1984), Princeton: Princeton University Press, Princeton · Zbl 0553.14020 |
| [17] | Kirwan, F., Symplectic implosion and nonreductive quotients, Geometric Aspects of Analysis and Mechanics, 213-256 (2011), Boston: Birkhäuser, Boston · Zbl 1256.14048 · doi:10.1007/978-0-8176-8244-6_9 |
| [18] | La Nave, G.; Tian, G., Soliton-type metrics and Kähler-Ricci flow on symplectic quotients, J Reine Angew Math, 711, 139-166 (2016) · Zbl 1339.53069 |
| [19] | Marsden, J.; Weinstein, A., Reduction of symplectic manifolds with symmetry, Rep Math Phys, 5, 121-130 (1974) · Zbl 0327.58005 · doi:10.1016/0034-4877(74)90021-4 |
| [20] | Mumford, D.; Fogarty, J.; Kirwan, J., Geometric Invariant Theory (1994), Berlin: Springer-Verlag, Berlin · Zbl 0797.14004 · doi:10.1007/978-3-642-57916-5 |
| [21] | Mundet i. Riera, I., A Hilbert-Mumford criterion for polystability in Kaehler geometry, Trans Amer Math Soc, 362, 5169-5187 (2010) · Zbl 1201.53086 · doi:10.1090/S0002-9947-2010-04831-7 |
| [22] | Ness, L., Mumford’s numerical function and stable projective hypersurfaces, Algebraic Geometry, 417-453 (1979), Berlin: Springer, Berlin · Zbl 0442.14019 · doi:10.1007/BFb0066655 |
| [23] | Ness, L., A stratification of the null cone via the moment map, Amer J Math, 106, 1281-1329 (1984) · Zbl 0604.14006 · doi:10.2307/2374395 |
| [24] | Ressayre, N., The GIT-equivalence for G-line bundles, Geom Dedicata, 81, 295-324 (2000) · Zbl 0955.14035 · doi:10.1023/A:1005275524522 |
| [25] | Safronov, P., Symplectic implosion and the Grothendieck-Springer resolution, Transform Groups, 22, 767-792 (2017) · Zbl 1377.53099 · doi:10.1007/s00031-016-9398-1 |
| [26] | Schmitt, A., A simple proof for the finiteness of GIT-quotients, Proc Amer Math Soc, 131, 359-362 (2003) · Zbl 1008.14009 · doi:10.1090/S0002-9939-02-06599-1 |
| [27] | Sjamaar, R., Holomorphic slices, symplectic reduction and multiplicities of representations, Ann of Math (2), 141, 87-129 (1995) · Zbl 0827.32030 · doi:10.2307/2118628 |
| [28] | Sjamaar, R.; Lerman, E., Stratified symplectic spaces and reduction, Ann of Math (2), 134, 375-422 (1991) · Zbl 0759.58019 · doi:10.2307/2944350 |
| [29] | Teleman, A., Symplectic stability, analytic stability in non-algebraic complex geometry, Internat J Math, 15, 183-209 (2004) · Zbl 1089.53058 · doi:10.1142/S0129167X04002223 |
| [30] | Thaddeus, M., Geometric invariant theory and flips, J Amer Math Soc, 9, 691-723 (1996) · Zbl 0874.14042 · doi:10.1090/S0894-0347-96-00204-4 |
| [31] | Woodward, C., Moment maps and geometric invariant theory, Les cours du CIRM, 1, 55-98 (2010) · doi:10.5802/ccirm.4 |
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