Moment maps and geometric invariant theory

Woodward, Chris

Actions hamiltoniennes : invariants et classification. 6 – 10 avril 2009, Les cours du CIRM, no. 1 (2010), pp. 55-98.

[1] R. Abraham and J. Marsden. Foundations of Mechanics. Benjamin/Cummings, Reading, 1978.

[2] S. Agnihotri and C. Woodward. Eigenvalues of products of unitary matrices and quantum Schubert calculus. Math. Res. Lett., 5(6):817–836, 1998.

[3] Dmitri N. Akhiezer. Lie group actions in complex analysis. Aspects of Mathematics, E27. Friedr. Vieweg & Sohn, Braunschweig, 1995.

[4] 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.

[5] M. F. Atiyah. Convexity and commuting Hamiltonians. Bull. London Math. Soc., 14:1–15, 1982.

[6] M. F. Atiyah and R. Bott. A Lefschetz fixed point formula for elliptic complexes. II. Applications. Ann. of Math. (2), 88:451–491, 1968.

[7] M. F. Atiyah and R. Bott. The moment map and equivariant cohomology. Topology, 23(1):1–28, 1984.

[8] M. Audin. The Topology of Torus Actions on Symplectic Manifolds, volume 93 of Progress in Mathematics. Birkhäuser, Boston, 1991.

[9] Chris Beasley and Edward Witten. Non-abelian localization for Chern-Simons theory. J. Differential Geom., 70(2):183–323, 2005.

[10] P. Belkale. Local systems on ${\mathbb{P}}^1-S$ for $S$ a finite set. Compositio Math., 129(1):67–86, 2001.

[11] Prakash Belkale and Shrawan Kumar. Eigenvalue problem and a new product in cohomology of flag varieties. Invent. Math., 166(1):185–228, 2006.

[12] A. Berenstein and R. Sjamaar. Coadjoint orbits, moment polytopes, and the Hilbert-Mumford criterion. J. Amer. Math. Soc., 13(2):433–466 (electronic), 2000.

[13] A. Białynicki-Birula. Some theorems on actions of algebraic groups. Ann. of Math. (2), 98:480–497, 1973.

[14] 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.

[15] Raoul Bott. Homogeneous vector bundles. Ann. of Math. (2), 66:203–248, 1957.

[16] 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.

[17] M. Brion. Groupe de Picard et nombres caractéristiques des variétés sphériques. Duke Math. J., 58(2):397–424, 1989.

[18] M. Brion, D. Luna, and Th. Vust. Espaces homogènes sphériques. Invent. Math., 84:617–632, 1986.

[19] M. Brion and M. Vergne. Lattice points in simple polytopes. J. Amer. Math. Soc., 10:371–392, 1997.

[20] L. Bruasse and A. Teleman. Harder-Narasimhan filtrations and optimal destabilizing vectors in complex geometry. Ann. Inst. Fourier (Grenoble), 55(3):1017–1053, 2005.

[21] Ana 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.

[22] J. S. Carter, D. E. Flath, and M. Saito. The classical and quantum 6$j$-symbols. Princeton University Press, Princeton, NJ, 1995.

[23] David A. Cox. The homogeneous coordinate ring of a toric variety. J. Algebraic Geom., 4(1):17–50, 1995.

[24] T. Delzant. Hamiltoniens périodiques et images convexes de l’application moment. Bull. Soc. Math. France, 116:315–339, 1988.

[25] T. Delzant. Classification des actions Hamiltoniennes des groupes de rang $2$. Ann. Global Anal. Geom., 8(1):87–112, 1990.

[26] S. K. Donaldson and P. Kronheimer. The geometry of four-manifolds. Oxford Mathematical Monographs. Oxford University Press, New York, 1990.

[27] 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.

[28] Dorothee Feldmüller. Two-orbit varieties with smaller orbit of codimension two. Arch. Math. (Basel), 54(6):582–593, 1990.

[29] W. Fulton. Introduction to Toric Varieties, volume 131 of Annals of Mathematics Studies. Princeton University Press, Princeton, 1993.

[30] William Fulton. Eigenvalues, invariant factors, highest weights, and Schubert calculus. Bull. Amer. Math. Soc. (N.S.), 37(3):209–249 (electronic), 2000.

[31] Alexander 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.

[32] V. Guillemin and S. Sternberg. Convexity properties of the moment mapping. Invent. Math., 67:491–513, 1982.

[33] V. Guillemin and S. Sternberg. Geometric quantization and multiplicities of group representations. Invent. Math., 67:515–538, 1982.

[34] V. Guillemin and S. Sternberg. Homogeneous quantization and multiplicities of group representations. J. Funct. Anal., 47:344–380, 1982.

[35] V. Guillemin and S. Sternberg. Geometric Asymptotics, volume 14 of Mathematical Surveys and Monographs. Amer. Math. Soc., Providence, R. I., revised edition, 1990.

[36] V. Guillemin and S. Sternberg. Symplectic Techniques in Physics. Cambridge Univ. Press, Cambridge, 1990.

[37] 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].

[38] Victor Guillemin. Kaehler structures on toric varieties. J. Differential Geom., 40(2):285–309, 1994.

[39] Victor Guillemin and Reyer Sjamaar. Convexity theorems for varieties invariant under a Borel subgroup. Pure Appl. Math. Q., 2(3, part 1):637–653, 2006.

[40] Victor Guillemin and Shlomo Sternberg. Multiplicity-free spaces. J. Differential Geom., 19(1):31–56, 1984.

[41] 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.

[42] J.-C. Hausmann and A. Knutson. The cohomology ring of polygon spaces. Ann. Inst. Fourier (Grenoble), 48(1):281–321, 1998.

[43] P. Heinzner and F. Loose. Reduction of complex Hamiltonian $G$-spaces. Geom. Funct. Anal., 4(3):288–297, 1994.

[44] Peter Heinzner and Alan Huckleberry. Kählerian structures on symplectic reductions. In Complex analysis and algebraic geometry, pages 225–253. de Gruyter, Berlin, 2000.

[45] S. Helgason. Differential geometry, Lie groups, and symmetric spaces. Academic Press, New York, 1978.

[46] Wim H. Hesselink. Uniform instability in reductive groups. J. Reine Angew. Math., 303/304:74–96, 1978.

[47] Wim H. Hesselink. Desingularizations of varieties of nullforms. Invent. Math., 55(2):141–163, 1979.

[48] A. Horn. Doubly stochastic matrices and the diagonal of a rotation matrix. Amer. J. Math., 76:620–630, 1954.

[49] A. Horn. Eigenvalues of sums of Hermitian matrices. Pacific J. Math., 12:225–241, 1962.

[50] Ignasi Mundet i Riera. A hilbert–mumford criterion for polystability in kaehler geometry, 2008. arXiv.org:0804.1067.

[51] L. C. Jeffrey and F. C. Kirwan. Localization for nonabelian group actions. Topology, 34:291–327, 1995.

[52] 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.

[53] F. C. Kirwan. Cohomology of Quotients in Symplectic and Algebraic Geometry, volume 31 of Mathematical Notes. Princeton Univ. Press, Princeton, 1984.

[54] F. C. Kirwan. Convexity properties of the moment mapping, III. Invent. Math., 77:547–552, 1984.

[55] 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.

[56] F. Knop. Automorphisms of multiplicity free hamiltonian manifolds. arXiv:1002.4256.

[57] Friedrich 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.

[58] 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.

[59] Allen Knutson and Terence Tao. Honeycombs and sums of Hermitian matrices. Notices Amer. Math. Soc., 48(2):175–186, 2001.

[60] Allen Knutson, Terence Tao, and Christopher 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.

[61] Allen Knutson, Terence Tao, and Christopher 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.

[62] 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.

[63] Bertram Kostant. On convexity, the Weyl group and the Iwasawa decomposition. Ann. Sci. École Norm. Sup. (4), 6:413–455 (1974), 1973.

[64] E. Lerman. Symplectic cuts. Math. Res. Letters, 2:247–258, 1995.

[65] E. Lerman, E. Meinrenken, S. Tolman, and C. Woodward. Non-abelian convexity by symplectic cuts. Topology, 37:245–259, 1998.

[66] Eugene Lerman. Gradient flow of the norm squared of a moment map. Enseign. Math. (2), 51(1-2):117–127, 2005.

[67] I. Losev. Proof of the Knop Conjecture. arXiv:math/0612561.

[68] D. Luna. Slices étales. Sur les groupes algébriques, Mém. Soc. Math. France, 33:81–105, 1973.

[69] D. Luna and Th. Vust. Plongements d’espaces homogènes. Comment. Math. Helv., 58(2):186–245, 1983.

[70] I. G. Macdonald. Symmetric functions and Hall polynomials. Oxford University Press, New York, 1995. With contributions by A. Zelevinsky.

[71] J. Marsden and A. Weinstein. Reduction of symplectic manifolds with symmetry. Rep. Math. Phys., 5:121–130, 1974.

[72] E. Meinrenken. Symplectic surgery and the Spin${}^{\mathrm{c}}$-Dirac operator. Adv. in Math., 134:240–277, 1998.

[73] K. Meyer. Symmetries and integrals in mathematics. In M. M. Peixoto, editor, Dynamical Systems, Univ. of Bahia, 1971, 1973. Academic Press, New York.

[74] 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.

[75] 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.

[76] 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.

[77] 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.

[78] 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.

[79] Tadao 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.

[80] P.-E. Paradan. The moment map and equivariant cohomology with generalized coefficients. Topology, 39(2):401–444, 2000.

[81] P.-E. Paradan. Localization of the Riemann-Roch character. J. Funct. Anal., 187(2):442–509, 2001.

[82] S. Ramanan and A. Ramanathan. Some remarks on the instability flag. Tohoku Math. J. (2), 36(2):269–291, 1984.

[83] N. Ressayre. Geometric invariant theory and generalized eigenvalue problem. arXiv:0704.2127.

[84] J. Roberts. Asymptotics and 6j-symbols. Geom. Topol. Monogr., 4:245–261, 2002. math.QA/0201177.

[85] Alexander H. W. Schmitt. Geometric invariant theory and decorated principal bundles. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2008.

[86] I. Schur. über eine klasse von mittelbindungen mit anwendungen auf der determinanten theorie. S. B. Berlin Math. Ges., 22:9–20, 1923.

[87] Jean-Pierre 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.

[88] 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.

[89] Stephen S. Shatz. The decomposition and specialization of algebraic families of vector bundles. Compositio Math., 35(2):163–187, 1977.

[90] G. C. Shephard. An elementary proof of Gram’s theorem for convex polytopes. Canad. J. Math., 19:1214–1217, 1967.

[91] R. Sjamaar. Holomorphic slices, symplectic reduction and multiplicities of representations. Ann. of Math. (2), 141:87–129, 1995.

[92] R. Sjamaar and E. Lerman. Stratified symplectic spaces and reduction. Ann. of Math. (2), 134:375–422, 1991.

[93] Peter 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.

[94] Andrei Teleman. Symplectic stability, analytic stability in non-algebraic complex geometry. Internat. J. Math., 15(2):183–209, 2004.

[95] C. Teleman. The quantization conjecture revisited. Ann. of Math. (2), 152(1):1–43, 2000.

[96] 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.

[97] Gang Tian. On a set of polarized Kähler metrics on algebraic manifolds. J. Differential Geom., 32(1):99–130, 1990.

[98] Katrin Wehrheim and Chris T. Woodward. Functoriality for Lagrangian correspondences in Floer theory. arXiv:0708.2851.

[99] 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.

[100] E. Witten. Two-dimensional gauge theories revisited. J. Geom. Phys., 9:303–368, 1992.

[101] Edward Witten. Holomorphic Morse inequalities. In Algebraic and differential topology—global differential geometry, volume 70 of Teubner-Texte Math., pages 318–333. Teubner, Leipzig, 1984.

[102] C. Woodward. The classification of transversal multiplicity-free group actions. Ann. Global Anal. Geom., 14:3–42, 1996.

[103] C. Woodward. Multiplicity-free Hamiltonian actions need not be Kähler. Invent. Math., 131(2):311–319, 1998.

[104] Chris 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.

[105] Siye Wu. Equivariant holomorphic Morse inequalities. II. Torus and non-abelian group actions. J. Differential Geom., 51(3):401–429, 1999.