Mirković-Vilonen cycles and polytopes

Abstract

We give an explicit description of the Mirković-Vilonen cycles on the affine Grassmannian for arbitrary complex reductive groups. We also give a combinatorial characterization of the MV polytopes. We prove that a polytope is an MV polytope if and only if it is a lattice polytope whose defining hyperplanes are parallel to those of the Weyl polytopes and whose 2-faces are rank 2 MV polytopes. As an application, we give a bijection between Lusztig’s canonical basis and the set of MV polytopes.

  • [jared1] J. E. Anderson, "On Mirković and Vilonen’s Intersection Homology cycles for the Loop Grassmannian," PhD Thesis , Princeton University, 2000.
    @phdthesis{jared1,
      author={Anderson, Jared E.},
      TITLE={On Mirković and Vilonen's Intersection Homology cycles for the Loop Grassmannian},
      SCHOOL={Princeton University},
      YEAR={2000},
      }
  • [jared2] Go to document J. E. Anderson, "A polytope calculus for semisimple groups," Duke Math. J., vol. 116, iss. 3, pp. 567-588, 2003.
    @article {jared2, MRKEY = {1958098},
      AUTHOR = {Anderson, Jared E.},
      TITLE = {A polytope calculus for semisimple groups},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {116},
      YEAR = {2003},
      NUMBER = {3},
      PAGES = {567--588},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {20G05 (14L99)},
      MRNUMBER = {2004a:20047},
      MRREVIEWER = {Timo Neuvonen},
      DOI = {10.1215/S0012-7094-03-11636-1},
      ZBLNUMBER = {1064.20047},
      }
  • [jaredmisha] Go to document J. Anderson and M. Kogan, "Mirković-Vilonen cycles and polytopes in Type A," Internat. Math. Res. Not., iss. 12, pp. 561-591, 2004.
    @article {jaredmisha, MRKEY = {2040622},
      AUTHOR = {Anderson, Jared and Kogan, Mikhail},
      TITLE = {Mirković-{V}ilonen cycles and polytopes in {T}ype {A}},
      JOURNAL = {Internat. Math. Res. Not.},
      FJOURNAL = {International Mathematics Research Notices},
      YEAR = {2004},
      NUMBER = {12},
      PAGES = {561--591},
      ISSN = {1073-7928},
      MRCLASS = {14C25 (14M25)},
      MRNUMBER = {2005d:14013},
      MRREVIEWER = {Baohua Fu},
      DOI = {10.1155/S1073792804132765},
      ZBLNUMBER = {1082.14012},
      }
  • [jaredmisha2] J. Anderson and M. Kogan, "The algebra of Mirković-Vilonen cycles in type A," Pure Appl. Math. Q., vol. 2, iss. 4, part 2, pp. 1187-1215, 2006.
    @article {jaredmisha2, MRKEY = {2282418},
      AUTHOR = {Anderson, Jared and Kogan, Mikhail},
      TITLE = {The algebra of {M}irković-{V}ilonen cycles in type {A}},
      JOURNAL = {Pure Appl. Math. Q.},
      FJOURNAL = {Pure and Applied Mathematics Quarterly},
      VOLUME = {2},
      YEAR = {2006},
      NUMBER = {4, part 2},
      PAGES = {1187--1215},
      ISSN = {1558-8599},
      MRCLASS = {14L35 (14C25)},
      MRNUMBER = {2007j:14071},
      MRREVIEWER = {Dmitrii A. Timash{ë}v},
      ZBLNUMBER = {1110.14043},
      }
  • [BD] A. Beilinson and V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves.
    @misc{BD,
      author={Beilinson, A. and Drinfeld, V.},
      TITLE={Quantization of Hitchin's integrable system and Hecke eigensheaves},
      NOTE={{\tt http://www.math.uchicago.edu/\verb'~'arinkin/langlands/}},
      }
  • [BFZA] Go to document A. Berenstein, S. Fomin, and A. Zelevinsky, "Parametrizations of canonical bases and totally positive matrices," Adv. Math., vol. 122, iss. 1, pp. 49-149, 1996.
    @article {BFZA, MRKEY = {1405449},
      AUTHOR = {Berenstein, Arkady and Fomin, Sergey and Zelevinsky, Andrei},
      TITLE = {Parametrizations of canonical bases and totally positive matrices},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {122},
      YEAR = {1996},
      NUMBER = {1},
      PAGES = {49--149},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {17B37 (05E15 20G05)},
      MRNUMBER = {98j:17008},
      MRREVIEWER = {Kailash C. Misra},
      DOI = {10.1006/aima.1996.0057},
      ZBLNUMBER = {0966.17011},
      }
  • [BFZ2] Go to document A. Berenstein, S. Fomin, and A. Zelevinsky, "Cluster algebras. III. Upper bounds and double Bruhat cells," Duke Math. J., vol. 126, iss. 1, pp. 1-52, 2005.
    @article {BFZ2, MRKEY = {2110627},
      AUTHOR = {Berenstein, Arkady and Fomin, Sergey and Zelevinsky, Andrei},
      TITLE = {Cluster algebras. {III}. {U}pper bounds and double {B}ruhat cells},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {126},
      YEAR = {2005},
      NUMBER = {1},
      PAGES = {1--52},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {16S99 (05E15 14M17 22E46)},
      MRNUMBER = {2005i:16065},
      DOI = {10.1215/S0012-7094-04-12611-9},
      ZBLNUMBER = {1135.16013},
      }
  • [BZschub] Go to document A. Berenstein and A. Zelevinsky, "Total positivity in Schubert varieties," Comment. Math. Helv., vol. 72, iss. 1, pp. 128-166, 1997.
    @article {BZschub, MRKEY = {1456321},
      AUTHOR = {Berenstein, Arkady and Zelevinsky, Andrei},
      TITLE = {Total positivity in {S}chubert varieties},
      JOURNAL = {Comment. Math. Helv.},
      FJOURNAL = {Commentarii Mathematici Helvetici},
      VOLUME = {72},
      YEAR = {1997},
      NUMBER = {1},
      PAGES = {128--166},
      ISSN = {0010-2571},
      CODEN = {COMHAX},
      MRCLASS = {14M15 (22E46)},
      MRNUMBER = {99g:14064},
      MRREVIEWER = {Kailash C. Misra},
      DOI = {10.1007/PL00000363},
      ZBLNUMBER = {0891.20030},
      }
  • [BZtpm] Go to document A. Berenstein and A. Zelevinsky, "Tensor product multiplicities, canonical and totally positive varieties," Invent. Math., vol. 143, iss. 1, pp. 77-128, 2001.
    @article {BZtpm, MRKEY = {1802793},
      AUTHOR = {Berenstein, Arkady and Zelevinsky, Andrei},
      TITLE = {Tensor product multiplicities, canonical and totally positive varieties},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {143},
      YEAR = {2001},
      NUMBER = {1},
      PAGES = {77--128},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {17B10 (12K10 17B20 17B37 20G05)},
      MRNUMBER = {2002c:17005},
      MRREVIEWER = {Ivan V. Arzhantsev},
      ZBLNUMBER = {1061.17006},
      DOI = {10.1007/s002220000102},
      }
  • [BG] Go to document A. Braverman and D. Gaitsgory, "Crystals via the affine Grassmannian," Duke Math. J., vol. 107, iss. 3, pp. 561-575, 2001.
    @article {BG, MRKEY = {1828302},
      AUTHOR = {Braverman, Alexander and Gaitsgory, Dennis},
      TITLE = {Crystals via the affine {G}rassmannian},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {107},
      YEAR = {2001},
      NUMBER = {3},
      PAGES = {561--575},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {20G05 (14F05)},
      MRNUMBER = {2002e:20083},
      MRREVIEWER = {Andrew Mathas},
      DOI = {10.1215/S0012-7094-01-10736-9},
      ZBLNUMBER = {1015.20030},
      }
  • [BFG] A. Braverman, M. Finkelberg, and D. Gaitsgory, "Uhlenbeck spaces via affine Lie algebras," in The Unity of Mathematics, Boston, MA: Birkhäuser, 2006, pp. 17-135.
    @incollection {BFG, MRKEY = {2181803},
      AUTHOR = {Braverman, Alexander and Finkelberg, Michael and Gaitsgory, Dennis},
      TITLE = {Uhlenbeck spaces via affine {L}ie algebras},
      BOOKTITLE = {The Unity of Mathematics},
      SERIES = {Progr. Math.},
      NUMBER = {244},
      PAGES = {17--135},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {2006},
      MRCLASS = {14D20 (17B67)},
      MRNUMBER = {2007f:14008},
      MRREVIEWER = {Ivan S. Kausz},
      ZBLNUMBER = {1105.14013},
      }
  • [E] G. Ewald, Combinatorial Convexity and Algebraic Geometry, New York: Springer-Verlag, 1996.
    @book {E, MRKEY = {1418400},
      AUTHOR = {Ewald, G{ü}nter},
      TITLE = {Combinatorial Convexity and Algebraic Geometry},
      SERIES = {Grad. Texts Math.},
      NUMBER = {168},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1996},
      PAGES = {xiv+372},
      ISBN = {0-387-94755-8},
      MRCLASS = {52B20 (14M25 52B05)},
      MRNUMBER = {97i:52012},
      MRREVIEWER = {P. McMullen},
      ZBLNUMBER = {0869.52001},
      }
  • [FM] M. Finkelberg and I. Mirković, "Semi-infinite flags. I. Case of global curve $\Bbb P^1$," in Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Providence, RI: Amer. Math. Soc., 1999, pp. 81-112.
    @incollection {FM, MRKEY = {1729360},
      AUTHOR = {Finkelberg, Michael and Mirkovi{ć},
      Ivan},
      TITLE = {Semi-infinite flags. {I}. {C}ase of global curve {$\bold P\sp 1$}},
      BOOKTITLE = {Differential Topology, Infinite-Dimensional {L}ie Algebras, and Applications},
      SERIES = {Amer. Math. Soc. Transl. Ser. 2},
      NUMBER = {194},
      PAGES = {81--112},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1999},
      MRCLASS = {14F43 (14M15 20G10 20G42)},
      MRNUMBER = {2001j:14029},
      MRREVIEWER = {Alexander E. Polishchuk},
      ZBLNUMBER = {1076.14512},
      }
  • [BF] E. Frenkel and D. Ben-Zvi, Vertex Algebras and Algebraic Curves, Second ed., Providence, RI: Amer. Math. Soc., 2004.
    @book {BF, MRKEY = {2082709},
      AUTHOR = {Frenkel, Edward and Ben-Zvi, David},
      TITLE = {Vertex Algebras and Algebraic Curves},
      SERIES = {Math. Surveys Monogr.},
      NUMBER = {88},
      EDITION = {Second},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2004},
      PAGES = {xiv+400},
      ISBN = {0-8218-3674-9},
      MRCLASS = {17B69 (14D21 14H81)},
      MRNUMBER = {2005d:17035},
      ZBLNUMBER = {1106.17035},
      }
  • [FZ] Go to document S. Fomin and A. Zelevinsky, "Double Bruhat cells and total positivity," J. Amer. Math. Soc., vol. 12, iss. 2, pp. 335-380, 1999.
    @article {FZ, MRKEY = {1652878},
      AUTHOR = {Fomin, Sergey and Zelevinsky, Andrei},
      TITLE = {Double {B}ruhat cells and total positivity},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {12},
      YEAR = {1999},
      NUMBER = {2},
      PAGES = {335--380},
      ISSN = {0894-0347},
      MRCLASS = {20G20 (15A23)},
      MRNUMBER = {2001f:20097},
      DOI = {10.1090/S0894-0347-99-00295-7},
      ZBLNUMBER = {0913.22011},
      }
  • [F] W. Fulton, Introduction to Toric Varieties, Princeton, NJ: Princeton Univ. Press, 1993.
    @book {F, MRKEY = {1234037},
      AUTHOR = {Fulton, William},
      TITLE = {Introduction to Toric Varieties},
      SERIES = {Ann. of Math. Studies},
      NUMBER = {131},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, NJ},
      YEAR = {1993},
      PAGES = {xii+157},
      ISBN = {0-691-00049-2},
      MRCLASS = {14M25 (14-02 14J30)},
      MRNUMBER = {94g:14028},
      MRREVIEWER = {T. Oda},
      ZBLNUMBER = {0813.14039},
      }
  • [GL] Go to document S. Gaussent and P. Littelmann, "LS galleries, the path model, and MV cycles," Duke Math. J., vol. 127, iss. 1, pp. 35-88, 2005.
    @article {GL, MRKEY = {2126496},
      AUTHOR = {Gaussent, S. and Littelmann, P.},
      TITLE = {L{S} galleries, the path model, and {MV} cycles},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {127},
      YEAR = {2005},
      NUMBER = {1},
      PAGES = {35--88},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {20G05 (14M15 17B10 22E46 51E24)},
      MRNUMBER = {2006c:20092},
      MRREVIEWER = {Guy Rousseau},
      DOI = {10.1215/S0012-7094-04-12712-5},
      ZBLNUMBER = {1078.22007},
      }
  • [G] V. Ginzburg, Perverse sheaves on a loop group and Langlands duality.
    @misc{G,
      author={Ginzburg, V.},
      TITLE={Perverse sheaves on a loop group and Langlands duality},
      NOTE={available at math.AG/9511007},
      }
  • [MVcrystal] Go to document J. Kamnitzer, "The crystal structure on the set of Mirković-Vilonen polytopes," Adv. Math., vol. 215, iss. 1, pp. 66-93, 2007.
    @article {MVcrystal, MRKEY = {2354986},
      AUTHOR = {Kamnitzer, Joel},
      TITLE = {The crystal structure on the set of {M}irković-{V}ilonen polytopes},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {215},
      YEAR = {2007},
      NUMBER = {1},
      PAGES = {66--93},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {17B37},
      MRNUMBER = {2009a:17021},
      MRREVIEWER = {Jae-Hoon Kwon},
      DOI = {10.1016/j.aim.2007.03.012},
      ZBLNUMBER = {1134.14028},
      }
  • [L] G. Lusztig, "Singularities, character formulas, and a $q$-analog of weight multiplicities," in Analysis and Topology on Singular Spaces, II, III (Luminy, 1981), Paris: Soc. Math. France, 1983, pp. 208-229.
    @incollection {L, MRKEY = {737932},
      AUTHOR = {Lusztig, George},
      TITLE = {Singularities, character formulas, and a {$q$}-analog of weight multiplicities},
      BOOKTITLE = {Analysis and Topology on Singular Spaces, {\rm II, III} ({L}uminy, 1981)},
      SERIES = {Astérisque},
      NUMBER = {101},
      PAGES = {208--229},
      PUBLISHER = {Soc. Math. France},
      ADDRESS = {Paris},
      YEAR = {1983},
      MRCLASS = {17B10 (05A30 20G05 22E47)},
      MRNUMBER = {85m:17005},
      MRREVIEWER = {James E. Humphreys},
      ZBLNUMBER = {0561.22013},
      }
  • [Lbook] Go to document G. Lusztig, "Canonical bases arising from quantized enveloping algebras," J. Amer. Math. Soc., vol. 3, iss. 2, pp. 447-498, 1990.
    @article {Lbook, MRKEY = {1035415},
      AUTHOR = {Lusztig, George},
      TITLE = {Canonical bases arising from quantized enveloping algebras},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {3},
      YEAR = {1990},
      NUMBER = {2},
      PAGES = {447--498},
      ISSN = {0894-0347},
      MRCLASS = {17B35 (16A64)},
      MRNUMBER = {90m:17023},
      MRREVIEWER = {Ya. S. So{\u\i}bel{\cprime}man},
      DOI = {10.2307/1990961},
      ZBLNUMBER = {0703.17008},
      }
  • [Lproj] G. Lusztig, "Introduction to quantized enveloping algebras," in New Developments in Lie Theory and their Applications (Córdoba, 1989), Boston, MA: Birkhäuser, 1992, pp. 49-65.
    @incollection {Lproj, MRKEY = {1190735},
      AUTHOR = {Lusztig, George},
      TITLE = {Introduction to quantized enveloping algebras},
      BOOKTITLE = {New Developments in {L}ie Theory and their Applications ({C}órdoba, 1989)},
      SERIES = {Progr. Math.},
      NUMBER = {105},
      PAGES = {49--65},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {1992},
      MRCLASS = {17B37 (17B10 20F36 81R50)},
      MRNUMBER = {94b:17029},
      MRREVIEWER = {Seok-Jin Kang},
      ZBLNUMBER = {0767.17014},
      }
  • [Lbookbook] G. Lusztig, Introduction to Quantum Groups, Boston, MA: Birkhäuser, 1993.
    @book {Lbookbook, MRKEY = {1227098},
      AUTHOR = {Lusztig, George},
      TITLE = {Introduction to Quantum Groups},
      SERIES = {Progr. Math.},
      NUMBER = {110},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {1993},
      PAGES = {xii+341},
      ISBN = {0-8176-3712-5},
      MRCLASS = {17B37 (16W30 17-02 17B35 81R50)},
      MRNUMBER = {94m:17016},
      MRREVIEWER = {Jie Du},
      ZBLNUMBER = {0788.17010},
      }
  • [Lfunnypaper] Go to document G. Lusztig, "An algebraic-geometric parametrization of the canonical basis," Adv. Math., vol. 120, iss. 1, pp. 173-190, 1996.
    @article {Lfunnypaper, MRKEY = {1392278},
      AUTHOR = {Lusztig, George},
      TITLE = {An algebraic-geometric parametrization of the canonical basis},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {120},
      YEAR = {1996},
      NUMBER = {1},
      PAGES = {173--190},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {17B37 (17B10 17B20)},
      MRNUMBER = {97g:17015},
      MRREVIEWER = {Kailash C. Misra},
      DOI = {10.1006/aima.1996.0036},
      ZBLNUMBER = {0877.17006},
      }
  • [MV] I. Mirković and K. Vilonen, "Perverse sheaves on affine Grassmannians and Langlands duality," Math. Res. Lett., vol. 7, iss. 1, pp. 13-24, 2000.
    @article {MV, MRKEY = {1748284},
      AUTHOR = {Mirkovi{ć},
      Ivan and Vilonen, Kari},
      TITLE = {Perverse sheaves on affine {G}rassmannians and {L}anglands duality},
      JOURNAL = {Math. Res. Lett.},
      FJOURNAL = {Mathematical Research Letters},
      VOLUME = {7},
      YEAR = {2000},
      NUMBER = {1},
      PAGES = {13--24},
      ISSN = {1073-2780},
      MRCLASS = {14F43 (20G05 32S60)},
      MRNUMBER = {2001h:14020},
      MRREVIEWER = {Alexander E. Polishchuk},
      ZBLNUMBER = {0987.14015},
      }
  • [MV2] Go to document I. Mirković and K. Vilonen, "Geometric Langlands duality and representations of algebraic groups over commutative rings," Ann. of Math., vol. 166, iss. 1, pp. 95-143, 2007.
    @article {MV2, MRKEY = {2342692},
      AUTHOR = {Mirkovi{ć},
      Ivan and Vilonen, Kari},
      TITLE = {Geometric {L}anglands duality and representations of algebraic groups over commutative rings},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {166},
      YEAR = {2007},
      NUMBER = {1},
      PAGES = {95--143},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {22E55 (11R39 20G05)},
      MRNUMBER = {2008m:22027},
      MRREVIEWER = {Peter Fiebig},
      DOI = {10.4007/annals.2007.166.95},
      ZBLNUMBER = {1138.22013},
      }
  • [speyer] Go to document D. E. Speyer, "Horn’s problem, Vinnikov curves, and the hive cone," Duke Math. J., vol. 127, iss. 3, pp. 395-427, 2005.
    @article {speyer, MRKEY = {2132865},
      AUTHOR = {Speyer, David E.},
      TITLE = {Horn's problem, {V}innikov curves, and the hive cone},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {127},
      YEAR = {2005},
      NUMBER = {3},
      PAGES = {395--427},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {14P99 (05E15 15A42)},
      MRNUMBER = {2005m:14113},
      MRREVIEWER = {Thomas C. Craven},
      DOI = {10.1215/S0012-7094-04-12731-0},
      ZBLNUMBER = {1069.14037},
      }
  • [SS] Go to document D. Speyer and B. Sturmfels, "The tropical Grassmannian," Adv. Geom., vol. 4, iss. 3, pp. 389-411, 2004.
    @article {SS, MRKEY = {2071813},
      AUTHOR = {Speyer, David and Sturmfels, Bernd},
      TITLE = {The tropical {G}rassmannian},
      JOURNAL = {Adv. Geom.},
      FJOURNAL = {Advances in Geometry},
      VOLUME = {4},
      YEAR = {2004},
      NUMBER = {3},
      PAGES = {389--411},
      ISSN = {1615-715X},
      CODEN = {AGDEA3},
      MRCLASS = {14P99 (13J30 14M25)},
      MRNUMBER = {2005d:14089},
      MRREVIEWER = {Meirav Amram-Blei},
      DOI = {10.1515/advg.2004.023},
      ZBLNUMBER = {1065.14071},
      }
  • [Z] G. M. Ziegler, Lectures on Polytopes, New York: Springer-Verlag, 1995.
    @book {Z, MRKEY = {1311028},
      AUTHOR = {Ziegler, G{ü}nter M.},
      TITLE = {Lectures on Polytopes},
      SERIES = {Grad. Texts Math.},
      NUMBER = {152},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1995},
      PAGES = {x+370},
      ISBN = {0-387-94365-X},
      MRCLASS = {52Bxx},
      MRNUMBER = {96a:52011},
      MRREVIEWER = {Margaret M. Bayer},
      ZBLNUMBER = {0823.52002},
      }

Authors

Joel Kamnitzer

Department of Mathematics, University of Toronto, Room 6290, 40 St. George Street, Toronto, Ontario, Canada M5S 2E4