Journal of Algebraic Geometry

A canonical decomposition of generalized theta functions on the moduli stack of Gieseker vector bundles

Author(s): Ivan Kausz
Journal: J. Algebraic Geom. 14 (2005), 439-480.
Posted: March 30, 2005
Retrieve article in: PDF DVI PostScript

Abstract: We use a Gieseker type degeneration of the moduli stack of vector bundles on a curve to prove a decomposition formula for generalized theta functions which is motivated by what in conformal field theory is called the factorization rule.

[B]: A. Beauville: Conformal blocks, fusion rules and the Verlinde formula. Israel Math. Conf. Proc. 9 (1996), 75-96. MR 1360497 (97f:17025)
[BL]: A. Beauville, Y. Laszlo: Conformal blocks and generalized theta functions. Comm. Math. Phys. 164 (1994), no. 2, 385-419. MR 1289330 (95k:14011)
[CP]: C. De Concini and C. Procesi: Complete Symmetric Varieties, Invariant theory (Montecatini, 1982), 1-44, Lecture Notes in Math., 996, Springer, Berlin-New York, 1983. MR 0718125 (85e:14070)
[DT]: R. Donagi and L. W. Tu: Theta functions for SL versus GL. Mathematical Research Letters 1, 345-357 (1994). MR 1302649 (95j:14012)
[EGA]: A. Grothendieck, worked out in collaboration with J. Dieudonné: Éléments de géométrie algébrique. Springer Verlag (1971) and Publications Mathématiques de l'I.H.E.S. No 8 (1961), 17 (1963), 20 (1964), 24 (1965), 28 (1966), 32 (1967). MR 0217084 (36:177b); MR 0163911 (29:1210); MR 0173675 (30:3885); MR 0199181 (33:7330); MR 0217086 (36:178); MR 0238860 (39:220)
[F1]: G. Faltings: A proof for the Verlinde formula. J. Algebraic Geom. 3 (1994), no. 2, 347-374. MR 1257326 (95j:14013)
[F2]: G. Faltings: Moduli-stacks for bundles on semistable curves. Math. Ann. 304 (1996), no. 3, 489-515. MR 1375622 (97d:14016)
[F3]: G. Faltings: Finiteness of coherent cohomology for proper fppf stacks. J. Algebraic Geometry 12 (2003) 357-366. MR 1949648 (2004e:14007)
[G]: D. Gieseker: A degeneration of the moduli space of stable bundles. J. Differential Geom. 19 (1984), no. 1, 173-206. MR 0739786 (85j:14014)
[Kn]: F. F. Knudsen: The projectivity of the moduli space of stable curves, II: The stacks $M_{g,n}$ . Math. Scand. 52 (1983), 161-199. MR 0702953 (85d:14038a)
[K1]: I. Kausz: A Modular Compactification of the General Linear Group, Documenta Math. 5 (2000) 553-594. MR 1796449 (2002a:14056)
[K2]: I. Kausz: A Gieseker Type Degeneration of Moduli Stacks of Vector Bundles on Curves, math.AG/0201197. To appear in Transactions of the AMS.
[K3]: I. Kausz: Global Sections of Line Bundles on a Wonderful Compactification of the General Linear Group. Preprint, 2003.
[LM]: G. Laumon and L. Moret-Bailly: Champs algébriques. Ergebnisse der Mathematik und ihrer Grenszgebiete, 3. Folge. Volume 39. Springer Verlag (2000). MR 1771927 (2001f:14006)
[LS]: Y. Laszlo, Ch. Sorger: The line bundles on the moduli of parabolic -bundles over curves and their sections. Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 4, 499-525. MR 1456243 (98f:14007)
[NR]: M.S. Narasimhan,T.R. Ramadas: Factorisation of generalised theta functions I. Invent. Math. 114 (1993), no. 3, 565-623. MR 1244913 (94i:14017)
[NS]: D.S. Nagaraj and C.S. Seshadri: Degenerations of the moduli spaces of vector bundles on curves II. Proc. Indian Acad. Sci. Math. Sci. 109 (1999), no 2, 165-201. MR 1687729 (2000c:14046)
[N]: P. E. Newstead: Introduction to moduli problems and orbit spaces. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 51 (1978). MR 0546290 (81k:14002)
[P]: C. Pauly: Espaces de modules de fibrés paraboliques et blocs conformes. Duke Math. J. 84 (1996), no. 1, 217-235. MR 1394754 (97h:14022)
[R]: T.R. Ramadas: Factorisation of generalised theta functions. II. The Verlinde formula. Topology 35 (1996), no. 3, 641-654. MR 1396770 (97j:14014)
[Se1]: C.S. Seshadri: Fibrés vectoriels sur les courbes algébriques. Notes written by J.-M. Drezet from a course at the École Normale Supérieure, June 1980. Astérisque, 96. Société Mathématique de France, Paris, 1982. MR 0699278 (85b:14023)
[Se2]: C.S. Seshadri: Degenerations of the moduli spaces of vector bundles on curves. School on Algebraic Geometry (Trieste, 1999), 205-265, ICTP Lect. Notes, 1, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2000. MR 1795864 (2001i:14044)
[SGA I]: A. Grothendieck: Revêtements étales et groupe fondamental (SGA 1). Springer Lecture Notes in Mathematics, Vol. 224. (1971). MR 0354651 (50:7129)
[So]: C. Sorger: La formule de Verlinde. Séminaire Bourbaki, Vol. 1994/95. Astérisque No. 237, (1996), Exp. No. 794, 3, 87-114. MR 1423621 (98f:14009)
[Sun1]: Xiaotao Sun: Degeneration of moduli spaces and generalized theta functions. J. Algebraic Geom. 9 (2000), no. 3, 459-527. MR 1752012 (2001h:14040)
[Sun2]: Xiaotao Sun: Degeneration of SL(n)-bundles on a reducible curve. Algebraic geometry in East Asia (Kyoto, 2001) 229-243 (math.AG/0112072). World Sci. Publishing, River Edge, NJ, 2002. MR 2030457 (2005a:14043)
[Sun3]: Xiaotao Sun: Moduli spaces of SL(r)-bundles on singular irreducible curves (math.AG/0303198). Asian J. Math. 7 (2003), 609-625. MR 2074894
[T]: C. Teleman: Borel-Weil-Bott theory on the moduli stack of -bundles over a curve. Invent. Math. 134 (1998), no. 1, 1-57. MR 1646586 (2000b:14014)
[TUY]: A. Tsuchiya, K. Ueno, Y. Yamada: Conformal field theory on universal family of stable curves with gauge symmetries. Integrable systems in quantum field theory and statistical mechanics, 459-566, Adv. Stud. Pure Math., 19, (1989). MR 1048605 (92a:81191)
[U]: K. Ueno: Introduction to conformal field theory with gauge symmetries. Geometry and physics (Aarhus, 1995), 603-745, Lecture Notes in Pure and Appl. Math., 184, (1997). MR 1423195 (98f:81306)

Previous issue \| Table of contents \| Next issue Recently posted articles \| Most recent issue \| All issues