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Geometric invariant theory and flips

Author: Michael Thaddeus
Journal: J. Amer. Math. Soc. 9 (1996), 691-723
MSC (1991): Primary 14L30, 14D20
MathSciNet review: 1333296
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Abstract: We study the dependence of geometric invariant theory quotients on the choice of a linearization. We show that, in good cases, two such quotients are related by a flip in the sense of Mori, and explain the relationship with the minimal model program. Moreover, we express the flip as the blow-up and blow-down of specific ideal sheaves, leading, under certain hypotheses, to a quite explicit description of the flip. We apply these ideas to various familiar moduli problems, recovering results of Kirwan, Boden-Hu, Bertram-Daskalopoulos-Wentworth, and the author. Along the way we display a chamber structure, following Duistermaat-Heckman, on the space of all linearizations. We also give a new, easy proof of the Bialynicki-Birula decomposition theorem.

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  • 1. M.F. Atiyah, On analytic surfaces with double points, Proc. R. Soc. Lond. A 247 (1958) 237--44. MR 20:2472
  • 2. A. Bertram, G. Daskalopoulos, and R. Wentworth, Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians, J. Amer. Math. Soc., to appear.
  • 3. U.N. Bhosle, Parabolic vector bundles on curves, Arkiv för Mat. 27 (1989) 15--22. MR 90f:14007
  • 4. A. Bialynicki-Birula, Some theorems on actions of algebraic groups, Ann. Math. 98 (1973) 480--497. MR 51:3186
  • 5. S. Bradlow, Special metrics and stability for holomorphic bundles with global sections, J. Diff. Geom. 33 (1991) 169--213. MR 91m:32031
  • 6. S. Bradlow and G. Daskalopoulos, Moduli of stable pairs for holomorphic bundles over Riemann surfaces, Int. J. Math. 2 (1991) 477-513. MR 93b:58026
  • 7. M. Brion and C. Procesi, Action d'un tore dans une variété projective, Operator algebras, unitary representations, enveloping algebras, and invariant theory, (A. Connes, M. Duflo, A. Joseph, and R. Rentschler, eds.), Birkhauser, 1990, pp. 509--539. MR 92m:14061
  • 8. H. Boden and Y. Hu, Variation of moduli of parabolic bundles, Math. Ann., to appear.
  • 9. I. Dolgachev and Y. Hu, Variation of geometric invariant theory quotients, preprint.
  • 10. J.-M. Drezet and M.S. Narasimhan, Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques, Inv. Math. 97 (1989) 53--94. MR 90d:14008
  • 11. J.J. Duistermaat and G.J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Inv. Math. 69 (1982) 259--268. Addendum: 72 (1983) 153--158. MR 84h:58051a,b
  • 12. D. Gieseker, On the moduli of vector bundles on an algebraic surface, Ann. of Math. 106 (1977) 45--60. MR 81h:14014
  • 13. A. Grassi and J. Kollár, Log canonical models, Astérisque 211 (1992) 29--45. MR 94f:14013
  • 14. A. Grothendieck, Sur quelques points d'algèbre homologique, Tôhoku Math. J. 9 (1957) 119--221. MR 21:1328
  • 15. A. Grothendieck, Technique de descente et théorèmes d'existence en géométrie algébrique, IV: Les schémas de Hilbert, Sém. Bourbaki 1960-61, Exp. 221; reprinted in Fondements de la géométrie algébrique, Secrétariat Math., Paris, 1962. MR 26:3566
  • 16. R. Hartshorne, Algebraic geometry, Springer-Verlag, 1977. MR 57:3116
  • 17. Y. Hu, The geometry and topology of quotient varieties of torus actions, Duke Math. J. 68 (1992) 151--184. Erratum: 68 (1992) 609. MR 93k:14019a,b
  • 18. F.C. Kirwan, Cohomology of quotients in algebraic and symplectic geometry, Princeton, 1984. MR 86i:58050
  • 19. D. Luna, Slices étales, Mém. Soc. Math. France 33 (1973) 81--105. MR 49:7269
  • 20. I.G. Macdonald, Symmetric products of an algebraic curve, Topology 1 (1962) 319--343. MR 27:1445
  • 21. V.B. Mehta and C.S. Seshadri, Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248 (1980) 205--239. MR 81i:14010
  • 22. S. Mori, Flip theorem and the existence of minimal models for 3-folds, J. Amer. Math. Soc. 1 (1988) 117--253. MR 89a:14048
  • 23. D. Mumford and J. Fogarty, Geometric invariant theory, second enlarged edition, Springer-Verlag, 1982. MR 86a:14006
  • 24. P.E. Newstead, Introduction to moduli problems and orbit spaces, Tata Inst., Bombay, 1978. MR 81k:14002
  • 25. C.S. Seshadri, Quotient spaces modulo reductive algebraic groups, Ann. of Math. 95 (1972) 511--556. MR 46:9044
  • 26. C.S. Seshadri, Fibrés vectoriels sur les courbes algébriques, Astérisque 96 (1982). MR 85b:14023
  • 27. M. Thaddeus, Stable pairs, linear systems and the Verlinde formula, Inv. Math. 117 (1994) 317--353. MR 95e:14006
  • 28. M. Thaddeus, Toric quotients and flips, Proceedings of the 1993 Taniguchi symposium, (K. Fukaya, M. Furuta, and T. Kohno, eds.), World Scientific, 1994, pp. 193--213. MR 96b:14067

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Additional Information

Michael Thaddeus
Affiliation: St. John’s College, Oxford, England
Address at time of publication: Department of Mathematics, Harvard University, 1 Oxford St., Cambridge, Massachusetts 02138

Received by editor(s): November 11, 1994
Received by editor(s) in revised form: March 23, 1995
Article copyright: © Copyright 1996 American Mathematical Society

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