Moment maps and geometric invariant theory—Corrected version (October 2011)

 Access to full text

 Full (PDF)

1. R. Abraham and J. Marsden. Foundations of Mechanics. Benjamin/Cummings, Reading, 1978. Zbl0393.70001MR515141
2. S. Agnihotri and C. Woodward. Eigenvalues of products of unitary matrices and quantum Schubert calculus. Math. Res. Lett., 5(6):817–836, 1998. Zbl1004.14013MR1671192
3. D. N. Akhiezer. Lie group actions in complex analysis. Aspects of Mathematics, E27. Friedr. Vieweg & Sohn, Braunschweig, 1995. Zbl0845.22001MR1334091
4. Werner Ballmann, Mikhael Gromov, and Viktor Schroeder. Manifolds of nonpositive curvature, volume 61 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 1985. Zbl0591.53001MR823981
5. B. H. Bowditch. A class of incomplete non-positively curved manifolds. Pacific J. Math., 172(1):1–39, 1996. Zbl0867.53039MR1379284
6. J. Arms, R. Cushman, and M. Gotay. A universal reduction procedure for Hamiltonian group actions. In T. Ratiu, editor, The Geometry of Hamiltonian Systems, volume 22 of Mathematical Sciences Research Institute Publications, Berkeley, 1989, 1991. Springer-Verlag, Berlin-Heidelberg-New York. Zbl0742.58016MR1123275
7. M. F. Atiyah. Convexity and commuting Hamiltonians. Bull. London Math. Soc., 14:1–15, 1982. Zbl0482.58013MR642416
8. M. F. Atiyah and R. Bott. A Lefschetz fixed point formula for elliptic complexes. II. Applications. Ann. of Math. (2), 88:451–491, 1968. Zbl0167.21703MR232406
9. M. F. Atiyah and R. Bott. The moment map and equivariant cohomology. Topology, 23(1):1–28, 1984. Zbl0521.58025MR721448
10. M. Audin. The Topology of Torus Actions on Symplectic Manifolds, volume 93 of Progress in Mathematics. Birkhäuser, Boston, 1991. Zbl0726.57029MR1106194
11. C. Beasley and E. Witten. Non-abelian localization for Chern-Simons theory. J. Differential Geom., 70(2):183–323, 2005. Zbl1097.58012MR2192257
12. P. Belkale. Local systems on ${\mathbb{P}}^1-S$ for $S$ a finite set. Compositio Math., 129(1):67–86, 2001. Zbl1042.14031MR1856023
13. P. Belkale and S. Kumar. Eigenvalue problem and a new product in cohomology of flag varieties. Invent. Math., 166(1):185–228, 2006. Zbl1106.14037MR2242637
14. A. Berenstein and R. Sjamaar. Coadjoint orbits, moment polytopes, and the Hilbert-Mumford criterion. J. Amer. Math. Soc., 13(2):433–466 (electronic), 2000. Zbl0979.53092MR1750957
15. A. Bialynicki-Birula. Some theorems on actions of algebraic groups. Ann. of Math. (2), 98:480–497, 1973. Zbl0275.14007MR366940
16. A. M. Bloch and T. S. Ratiu. Convexity and integrability. In Symplectic geometry and mathematical physics (Aix-en-Provence, 1990), volume 99 of Progr. Math., pages 48–79. Birkhäuser Boston, Boston, MA, 1991. Zbl0755.53023MR1156534
17. R. Bott. Homogeneous vector bundles. Ann. of Math. (2), 66:203–248, 1957. Zbl0094.35701MR89473
18. M. Brion. Sur l’image de l’application moment. In M.-P. Malliavin, editor, Séminaire d’algèbre Paul Dubreuil et Marie-Paule Malliavin, volume 1296 of Lecture Notes in Mathematics, pages 177–192, Paris, 1986, 1987. Springer-Verlag, Berlin-Heidelberg-New York. Zbl0667.58012MR932055
19. M. Brion. Groupe de Picard et nombres caractéristiques des variétés sphériques. Duke Math. J., 58(2):397–424, 1989. Zbl0701.14052MR1016427
20. M. Brion, D. Luna, and Th. Vust. Espaces homogènes sphériques. Invent. Math., 84:617–632, 1986. Zbl0604.14047MR837530
21. M. Brion and M. Vergne. Lattice points in simple polytopes. J. Amer. Math. Soc., 10:371–392, 1997. Zbl0871.52009MR1415319
22. L. Bruasse and A. Teleman. Harder-Narasimhan filtrations and optimal destabilizing vectors in complex geometry. Ann. Inst. Fourier (Grenoble), 55(3):1017–1053, 2005. Zbl1093.32009MR2149409
23. A. Cannas da Silva. Introduction to symplectic and Hamiltonian geometry. Publicações Matemáticas do IMPA. [IMPA Mathematical Publications]. Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2003. Zbl1073.53102MR2115646
24. Pierre-Emmanuel Caprace and Alexander Lytchak. At infinity of finite-dimensional CAT(0) spaces. Math. Ann., 346(1):1–21, 2010. Zbl1184.53038MR2558883
25. J. S. Carter, D. E. Flath, and M. Saito. The classical and quantum 6$j$-symbols. Princeton University Press, Princeton, NJ, 1995. Zbl0851.17001MR1366832
26. X. Chen and S. Sun. Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature Kähler metrics. arXiv:1004.2012. Zbl1307.53058
27. D. A. Cox. The homogeneous coordinate ring of a toric variety. J. Algebraic Geom., 4(1):17–50, 1995. Zbl0846.14032MR1299003
28. T. Delzant. Hamiltoniens périodiques et images convexes de l’application moment. Bull. Soc. Math. France, 116:315–339, 1988. Zbl0676.58029MR984900
29. T. Delzant. Classification des actions Hamiltoniennes des groupes de rang $2$. Ann. Global Anal. Geom., 8(1):87–112, 1990. Zbl0711.58017MR1075241
30. S. K. Donaldson and P. Kronheimer. The geometry of four-manifolds. Oxford Mathematical Monographs. Oxford University Press, New York, 1990. Zbl0904.57001MR1079726
31. J. J. Duistermaat. Equivariant cohomology and stationary phase. In Symplectic geometry and quantization, (Sanda and Yokohama, 1993), volume 179 of Contemp. Math., pages 45–62, Providence, RI, 1994. Amer. Math. Soc. Zbl0852.57029MR1319601
32. P. B. Eberlein. Geometry of nonpositively curved manifolds. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1996. Zbl0883.53003MR1441541
33. D. Feldmüller. Two-orbit varieties with smaller orbit of codimension two. Arch. Math. (Basel), 54(6):582–593, 1990. Zbl0668.14031MR1052980
34. W. Fulton. Introduction to Toric Varieties, volume 131 of Annals of Mathematics Studies. Princeton University Press, Princeton, 1993. Zbl0813.14039MR1234037
35. W. Fulton. Eigenvalues, invariant factors, highest weights, and Schubert calculus. Bull. Amer. Math. Soc. (N.S.), 37(3):209–249 (electronic), 2000. Zbl0994.15021MR1754641
36. A. Grothendieck. Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2). Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 4. Société Mathématique de France, Paris, 2005. Séminaire de Géométrie Algébrique du Bois Marie, 1962, Augmenté d’un exposé de Michèle Raynaud. [With an exposé by Michèle Raynaud], With a preface and edited by Yves Laszlo, Revised reprint of the 1968 French original. Zbl0197.47202
37. V. Guillemin and S. Sternberg. Convexity properties of the moment mapping. Invent. Math., 67:491–513, 1982. Zbl0503.58017MR664117
38. V. Guillemin and S. Sternberg. Geometric quantization and multiplicities of group representations. Invent. Math., 67:515–538, 1982. Zbl0503.58018MR664118
39. V. Guillemin and S. Sternberg. Homogeneous quantization and multiplicities of group representations. J. Funct. Anal., 47:344–380, 1982. Zbl0733.58021MR665022
40. V. Guillemin and S. Sternberg. Geometric Asymptotics, volume 14 of Mathematical Surveys and Monographs. Amer. Math. Soc., Providence, R. I., revised edition, 1990. Zbl0364.53011MR516965
41. V. Guillemin and S. Sternberg. Symplectic Techniques in Physics. Cambridge Univ. Press, Cambridge, 1990. Zbl0734.58005MR1066693
42. V. W. Guillemin and S. Sternberg. Supersymmetry and equivariant de Rham theory. Springer-Verlag, Berlin, 1999. With an appendix containing two reprints by Henri Cartan [MR 13,107e; MR 13,107f]. Zbl0934.55007MR1689252
43. V. Guillemin. Kaehler structures on toric varieties. J. Differential Geom., 40(2):285–309, 1994. Zbl0813.53042MR1293656
44. V. Guillemin and R. Sjamaar. Convexity theorems for varieties invariant under a Borel subgroup. Pure Appl. Math. Q., 2(3, part 1):637–653, 2006. Zbl1107.53055MR2252111
45. V. Guillemin and S. Sternberg. Multiplicity-free spaces. J. Differential Geom., 19(1):31–56, 1984. Zbl0548.58017MR739781
46. R. Hartshorne. Residues and duality. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20. Springer-Verlag, Berlin, 1966. MR222093
47. J.-C. Hausmann and A. Knutson. The cohomology ring of polygon spaces. Ann. Inst. Fourier (Grenoble), 48(1):281–321, 1998. Zbl0903.14019MR1614965
48. G. J. Heckman. Projections of orbits and asymptotic behavior of multiplicities for compact Lie groups. Invent. Math., 67:333–356, 1982. Zbl0497.22006MR665160
49. P. Heinzner and F. Loose. Reduction of complex Hamiltonian $G$-spaces. Geom. Funct. Anal., 4(3):288–297, 1994. Zbl0816.53018MR1274117
50. P. Heinzner and A. Huckleberry. Kählerian structures on symplectic reductions. In Complex analysis and algebraic geometry, pages 225–253. de Gruyter, Berlin, 2000. Zbl0999.32011MR1760879
51. S. Helgason. Differential geometry, Lie groups, and symmetric spaces. Academic Press, New York, 1978. Zbl0451.53038MR514561
52. Wim H. Hesselink. Uniform instability in reductive groups. J. Reine Angew. Math., 303/304:74–96, 1978. Zbl0386.20020MR514673
53. Wim H. Hesselink. Desingularizations of varieties of nullforms. Invent. Math., 55(2):141–163, 1979. Zbl0401.14006MR553706
54. A. Horn. Doubly stochastic matrices and the diagonal of a rotation matrix. Amer. J. Math., 76:620–630, 1954. Zbl0055.24601MR63336
55. A. Horn. Eigenvalues of sums of Hermitian matrices. Pacific J. Math., 12:225–241, 1962. Zbl0112.01501MR140521
56. Ignasi Mundet i Riera. A Hilbert–Mumford criterion for polystability in Kaehler geometry, 2008. arXiv.org:0804.1067. Zbl1201.53086MR2657676
57. L. C. Jeffrey and F. C. Kirwan. Localization for nonabelian group actions. Topology, 34:291–327, 1995. Zbl0833.55009MR1318878
58. V. A. Kaimanovich. Lyapunov exponents, symmetric spaces and a multiplicative ergodic theorem for semisimple Lie groups. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 164(Differentsialnaya Geom. Gruppy Li i Mekh. IX):29–46, 196–197, 1987. Zbl0696.22012MR947327
59. M. Kapovich, B. Leeb, and J. Millson. 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. Zbl1167.53044MR2472176
60. G. Kempf and L. Ness. The length of vectors in representation spaces. In K. Lonsted, editor, Algebraic Geometry, volume 732 of Lecture Notes in Mathematics, pages 233–244, Copenhagen, 1978, 1979. Springer-Verlag, Berlin-Heidelberg-New York. Zbl0407.22012MR555701
61. F. C. Kirwan. Cohomology of Quotients in Symplectic and Algebraic Geometry, volume 31 of Mathematical Notes. Princeton Univ. Press, Princeton, 1984. Zbl0553.14020MR766741
62. F. C. Kirwan. Convexity properties of the moment mapping, III. Invent. Math., 77:547–552, 1984. Zbl0561.58016MR759257
63. A. A. Klyachko. Equivariant vector bundles on toric varieties and some problems of linear algebra. In Topics in algebra, Part 2 (Warsaw, 1988), pages 345–355. PWN, Warsaw, 1990. Zbl0761.14017MR1171283
64. F. Knop. Automorphisms of multiplicity free Hamiltonian manifolds. arXiv:1002.4256. Zbl1226.53082MR2748401
65. F. Knop. The Luna-Vust theory of spherical embeddings. In Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), pages 225–249, Madras, 1991. Manoj Prakashan. Zbl0812.20023MR1131314
66. A. Knutson and T. Tao. The honeycomb model of ${\mathrm{g}\mathrm{l}}_n\left(\mathbf{c}\right)$ tensor products. I. Proof of the saturation conjecture. J. Amer. Math. Soc., 12(4):1055–1090, 1999. Zbl0944.05097MR1671451
67. A. Knutson and T. Tao. Honeycombs and sums of Hermitian matrices. Notices Amer. Math. Soc., 48(2):175–186, 2001. Zbl1047.15006MR1811121
68. A. Knutson, T. Tao, and C. Woodward. The honeycomb model of ${\mathrm{GL}}_n\left(ℂ\right)$ tensor products. II. Puzzles determine facets of the Littlewood-Richardson cone. J. Amer. Math. Soc., 17(1):19–48 (electronic), 2004. Zbl1043.05111MR2015329
69. A. Knutson, T. Tao, and C. Woodward. A positive proof of the Littlewood-Richardson rule using the octahedron recurrence. Electron. J. Combin., 11(1):Research Paper 61, 18 pp. (electronic), 2004. Zbl1053.05119MR2097327
70. B. Kostant. Quantization and unitary representations. In C. T. Taam, editor, Lectures in Modern Analysis and Applications III, volume 170 of Lecture Notes in Mathematics, pages 87–208, Washington, D.C., 1970. Springer-Verlag, Berlin-Heidelberg-New York. Zbl0223.53028MR294568
71. Bertram Kostant. On convexity, the Weyl group and the Iwasawa decomposition. Ann. Sci. École Norm. Sup. (4), 6:413–455 (1974), 1973. Zbl0293.22019MR364552
72. E. Lerman. Symplectic cuts. Math. Res. Letters, 2:247–258, 1995. Zbl0835.53034MR1338784
73. E. Lerman, E. Meinrenken, S. Tolman, and C. Woodward. Non-abelian convexity by symplectic cuts. Topology, 37:245–259, 1998. Zbl0913.58023MR1489203
74. E. Lerman. Gradient flow of the norm squared of a moment map. Enseign. Math. (2), 51(1-2):117–127, 2005. Zbl1103.53051MR2154623
75. I. V. Losev. Proof of the Knop conjecture. Ann. Inst. Fourier (Grenoble), 59(3):1105–1134, 2009. Zbl1191.14075MR2543664
76. D. Luna. Slices étales. Sur les groupes algébriques, Mém. Soc. Math. France, 33:81–105, 1973. Zbl0286.14014MR342523
77. D. Luna and Th. Vust. Plongements d’espaces homogènes. Comment. Math. Helv., 58(2):186–245, 1983. Zbl0545.14010MR705534
78. I. G. Macdonald. Symmetric functions and Hall polynomials. Oxford University Press, New York, 1995. With contributions by A. Zelevinsky. Zbl0824.05059MR1354144
79. J. Marsden and A. Weinstein. Reduction of symplectic manifolds with symmetry. Rep. Math. Phys., 5:121–130, 1974. Zbl0327.58005MR402819
80. E. Meinrenken. Symplectic surgery and the Spin${}^{\mathrm{c}}$-Dirac operator. Adv. in Math., 134:240–277, 1998. Zbl0929.53045MR1617809
81. K. Meyer. Symmetries and integrals in mathematics. In M. M. Peixoto, editor, Dynamical Systems, Univ. of Bahia, 1971, 1973. Academic Press, New York. MR331427
82. D. Mumford, J. Fogarty, and F. Kirwan. Geometric Invariant Theory, volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete, 2. Folge. Springer-Verlag, Berlin-Heidelberg-New York, third edition, 1994. Zbl0147.39304MR1304906
83. M. S. Narasimhan and C. S. Seshadri. Stable and unitary vector bundles on a compact Riemann surface. Ann. of Math. (2), 82:540–567, 1965. Zbl0171.04803MR184252
84. L. Ness. A stratification of the null cone via the moment map. Amer. J. Math., 106(6):1281–1329, 1984. with an appendix by D. Mumford. Zbl0604.14006MR765581
85. P. E. Newstead. Introduction to moduli problems and orbit spaces, volume 51 of Tata Institute of Fundamental Research Lectures on Mathematics and Physics. Tata Institute of Fundamental Research, Bombay, 1978. Zbl0411.14003MR546290
86. P. E. Newstead. Geometric invariant theory. In Moduli spaces and vector bundles, volume 359 of London Math. Soc. Lecture Note Ser., pages 99–127. Cambridge Univ. Press, Cambridge, 2009. Zbl1187.14054MR2537067
87. T. Oda. Convex bodies and algebraic geometry, volume 15 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties, Translated from the Japanese. Zbl0628.52002MR922894
88. P.-E. Paradan. The moment map and equivariant cohomology with generalized coefficients. Topology, 39(2):401–444, 2000. Zbl0941.37050MR1722000
89. P.-E. Paradan. Localization of the Riemann-Roch character. J. Funct. Anal., 187(2):442–509, 2001. Zbl1001.53062MR1875155
90. S. Ramanan and A. Ramanathan. Some remarks on the instability flag. Tohoku Math. J. (2), 36(2):269–291, 1984. Zbl0567.14027MR742599
91. N. Ressayre. Geometric invariant theory and generalized eigenvalue problem. arXiv:0704.2127. Zbl1197.14051MR2609246
92. J. Roberts. Asymptotics and 6j-symbols. Geom. Topol. Monogr., 4:245–261, 2002. math.QA/0201177. Zbl1012.22037MR2002614
93. A. H. W. Schmitt. Geometric invariant theory and decorated principal bundles. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2008. Zbl1159.14001MR2437660
94. I. Schur. über eine klasse von mittelbindungen mit anwendungen auf der determinanten theorie. S. B. Berlin Math. Ges., 22:9–20, 1923. 
95. J. -P. Serre. Représentations linéaires et espaces homogènes kählériens des groupes de Lie compacts (d’après Armand Borel et André Weil). In Séminaire Bourbaki, Vol. 2, pages Exp. No. 100, 447–454. Soc. Math. France, Paris, 1995. Zbl0121.16203MR1609256
96. C. S. Seshadri. Fibrés vectoriels sur les courbes algébriques, volume 96 of Astérisque. Société Mathématique de France, Paris, 1982. Notes written by J.-M. Drezet from a course at the École Normale Supérieure, June 1980. Zbl0517.14008MR699278
97. S. S. Shatz. The decomposition and specialization of algebraic families of vector bundles. Compositio Math., 35(2):163–187, 1977. Zbl0371.14010MR498573
98. G. C. Shephard. An elementary proof of Gram’s theorem for convex polytopes. Canad. J. Math., 19:1214–1217, 1967. Zbl0157.52504MR225228
99. R. Sjamaar. Holomorphic slices, symplectic reduction and multiplicities of representations. Ann. of Math. (2), 141:87–129, 1995. Zbl0827.32030MR1314032
100. R. Sjamaar and E. Lerman. Stratified symplectic spaces and reduction. Ann. of Math. (2), 134:375–422, 1991. Zbl0759.58019MR1127479
101. P. Slodowy. Die Theorie der optimalen Einparameteruntergruppen für instabile Vektoren. In Algebraische Transformationsgruppen und Invariantentheorie, volume 13 of DMV Sem., pages 115–131. Birkhäuser, Basel, 1989. Zbl0753.14006MR1044588
102. G. Székelyhidi. Extremal metrics and K-stability (PhD thesis). arxiv:math/0611002. MR2303522
103. A. Teleman. Symplectic stability, analytic stability in non-algebraic complex geometry. Internat. J. Math., 15(2):183–209, 2004. Zbl1089.53058MR2055369
104. C. Teleman. The quantization conjecture revisited. Ann. of Math. (2), 152(1):1–43, 2000. Zbl0980.53102MR1792291
105. R. P. Thomas. Notes on GIT and symplectic reduction for bundles and varieties. In Surveys in differential geometry. Vol. X, volume 10 of Surv. Differ. Geom., pages 221–273. Int. Press, Somerville, MA, 2006. Zbl1132.14043MR2408226
106. G. Tian. On a set of polarized Kähler metrics on algebraic manifolds. J. Differential Geom., 32(1):99–130, 1990. Zbl0706.53036MR1064867
107. Katrin Wehrheim and Chris T. Woodward. Functoriality for Lagrangian correspondences in Floer theory. Quantum Topol., 1:129–170, 2010. Zbl1206.53088MR2657646
108. A. Weinstein. The symplectic “category”. In Differential geometric methods in mathematical physics (Clausthal, 1980), volume 905 of Lecture Notes in Math., pages 45–51. Springer, Berlin, 1982. Zbl0486.58017MR657441
109. E. Witten. Two-dimensional gauge theories revisited. J. Geom. Phys., 9:303–368, 1992. Zbl0768.53042MR1185834
110. E. Witten. Holomorphic Morse inequalities. In Algebraic and differential topology—global differential geometry, volume 70 of Teubner-Texte Math., pages 318–333. Teubner, Leipzig, 1984. Zbl0588.32009MR792703
111. C. Woodward. The classification of transversal multiplicity-free group actions. Ann. Global Anal. Geom., 14:3–42, 1996. Zbl0877.58022MR1375064
112. C. Woodward. Multiplicity-free Hamiltonian actions need not be Kähler. Invent. Math., 131(2):311–319, 1998. Zbl0902.58014MR1608579
113. C. T. Woodward. Localization via the norm-square of the moment map and the two-dimensional Yang-Mills integral. J. Symp. Geom., 3(1):17–55, 2006. Zbl1103.53052MR2198772
114. S. Wu. Equivariant holomorphic Morse inequalities. II. Torus and non-abelian group actions. J. Differential Geom., 51(3):401–429, 1999. Zbl1024.58007MR1726735