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Compactifications of Moduli of G-Bundles and Conformal Blocks

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Abstract

For a simple Lie algebra of type A or C and a genus \(g\ge 2\), we show that the conformal blocks algebra on \(\overline{\mathcal {M}}_g\) is finitely generated and relate conformal blocks over singular curves to Schmitt and Muñoz-Castañeda’s compactification of the moduli space of G-bundles.

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Data Availability

All relevant data and materials that support the findings of this study are available from the corresponding author upon reasonable request.

Notes

  1. Here, and throughout, we make no distinction between a vector bundle and its sheaf of sections. If \(\mathcal {E}\) is a vector bundle, then its sheaf of sections will also be denoted \(\mathcal {E}\).

  2. Note that the definition of \(Q_G(\lambda )\) is somewhat “opposite” to the usual one—usually, it would be \(\lambda (t)g\lambda (t)^{-1}\) appearing in (2). This is because we follow Schmitt’s convention of listing weights in increasing order, meaning \(Q_G(\lambda )\) preserves a flag in V where the subquotients \(V_1/V_0,V_2/V_1,V_3/V_2,\dots \) have increasing \(\mathbb {G}_m\)-weights under \(\lambda \).

  3. Every group has such a representation, because if V is any representation, then \(V\oplus V^*\) has an invariant, nondegenerate bilinear form.

  4. Indeed, proposition 2.2 applies, because degree is constant in flat families.

  5. To see this, let \(\mathcal {O}(1)\) be an ample line bundle on \(C_0\) and choose a large n such that \(\text{ H}^i(C_0\times T,\mathscr {E}(n))=0\) for \(i>0\). Since \(\mathscr {E}\) is a flat famiy of torsion-free sheaves on a family of curves, there is a short exact sequence \(0\rightarrow \mathscr {E}\rightarrow \mathscr {E}(n)\rightarrow \mathscr {E}(n)/\mathscr {E}\rightarrow 0\), where \(\mathscr {E}(n)/\mathscr {E}\) has no higher cohomology because it is isomorphic to \(\left. \mathscr {E}(n)\right| _{D\times T}\) for a divisor \(D\subset C_0\) and \(D\times T\) is affine.

  6. By “\(Q_x\)” in Eq. (13), we really mean the skyscraper sheaf \((i_x)_*Q_x\) concentrated at x. We will continue to use this abuse of notation throughout the paper.

References

  1. Ramanathan, A.: Moduli of principal bundles over algebraic curves I-II. Proceedings of the Indian Academy of Science 106(301–328), 421–449 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Nagaraj, D.S., Seshadri, C.S.: Degenerations of the moduli spaces of vector bundles on curves, I. Proc. Indian Acad. Sci. Math. Sci. 107(2), 101–137 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. D. S. Nagaraj, C. S. Seshadri, Degenerations of the moduli spaces of vector bundles on curves. II. Generalized Gieseker moduli spaces, Proc. Indian Acad. Sci. Math. Sci. 109 (1999), no. 2, 165-201

  4. Pandharipande, R.: A compactification over \(\overline{M}_{g}\) of the universal moduli space of slope-semistable vector bundles. J. Amer. Math. Soc. 9(2), 425–471 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Seshadri, C.S.: Fibrés Vectoriels sur les Courbes Algébriques, Asterisque, 96. Société Matématique de France, Paris (1982)

    MATH  Google Scholar 

  6. X. Sun, Degenerations of SL(n)-bundles on a reducible curve, Proceedings of the Symposium on Algebraic Geometry in East Asia (2001), 3-10

  7. Faltings, G.: Moduli-stacks for bundles on semistable curves. Math. Ann. 304, 489–515 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. Schmitt, A.H.W.: Singular principal G-bundles on nodal curves. J. Eur. Math. Soc. 7(2), 215–251 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. A. H. W. Schmitt, Moduli spaces for semistable honest singular principal bundles on a nodal curve which are compatible with degeneration. A remark on U. N. Bhosle’s paper: “Tensor fields and singular principal bundles,” Int. Math. Res. Not. (2005), no. 23, 1427-1437

  10. A. L. Muñoz-Castañeda, A. H. W. Schmitt, Singular principal bundles on reducible nodal curves,http://arxiv.org/abs/1911.01578arXiv: 1911.01578 (2020)

  11. A. L. Muñoz-Castañeda, A compactification of the universal moduli space of principal G-bundles, Mediterr. J. Math. 19, No. 2, Paper No. 54, 23 p. (2022)

  12. Beauville, A., Laszlo, Y.: Conformal blocks and generalized theta functions. Comm. Math. Phys. 164(2), 385–419 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Belkale, P., Fakhruddin, N.: Triviality properties of principal bundles on singular curves. Algebr. Geom. 6(2), 234–259 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  14. Faltings, G.: A proof for the Verlinde formula. J. Algebraic Geom. 3(2), 347–374 (1994)

    MathSciNet  MATH  Google Scholar 

  15. Kumar, S., Narasimhan, M.S., Ramanathan, A.: Infinite Grassmannians and moduli spaces of G-bundles. Math. Ann. 300(1), 41–75 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  16. Y. Laszlo, C. Sorger, The line bundles on the moduli of parabolic G-bundles over curves and their sections, Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 4, 499-525

  17. Manon, C.: The algebra of conformal blocks. J. Eur. Math. Soc. 20(11), 2685–2715 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  18. Belkale, P., Gibney, A.: On finite generation of the section ring of the determinant of cohomology line bundle. Trans. Amer. Math. Soc. 371(10), 7199–7242 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  19. H. Moon, S. Yoo, Finite generation of the algebra of type A conformal blocks via birational geometry II: higher genus, Proc. Lond. Math. Soc. (3) 120 (2020), no. 2, 242-264

  20. 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, Adv. Stud. Pure Math., vol. 19, Academic Press, Boston, MA, 1989

  21. Sun, X.: Factorization of generalized theta functions in the reducible case. Ark. Mat. 41(1), 165–202 (2003)

    Article  MathSciNet  Google Scholar 

  22. V. Balaji, Torsors on semistable curves and degenerations, Proc. Indian Acad. Sci. Math. Sci. 132 (2022), no. 1, Paper No. 27, 63 pp

  23. P. Solis, A complete degeneration of the moduli of \(G\)-bundles on a curve, http://arxiv.org/abs/1311.6847

  24. Martens, J.: Group compactifications and moduli spaces, analytic and algebraic geometry, pp. 187–206. Springer & Hindustan Book Agency, Singapore (2017)

    Book  MATH  Google Scholar 

  25. Martens, J., Thaddeus, M.: Compactifications of reductive groups as moduli stacks of bundles. Compos. Math. 152(1), 62–98 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Baily, W.L., Jr.: Satake’s compactification of \(V_n\). Amer. J. Math. 80, 348–364 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  27. Faltings, G., Chai, C.: Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 22. Springer-Verlag, Berlin (1990)

    Book  Google Scholar 

  28. J. Wang, The moduli stack of G-bundles,http://arxiv.org/abs/1104.4828arXiv:1104.4828 (2011)

  29. A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. II, Inst. Hautes Études Sci. Publ. Math (1963), no. 17

  30. B. Fantechi, L. Göttsche, L. Illusie, S. L. Kleiman, N. Nitsure, A. Vistoli, Fundamental algebraic geometry. Grothendieck’s FGA Explained, Mathematical Surveys and Monographs, Vol. 123, American Mathematical Society, Providence, RI, 2005

  31. K. Ascher, D. Bejleri, Moduli of weighted stable elliptic surfaces and invariance of log plurigenera, Proc. Lond. Math. Soc. (3) 122, No. 5, 617-677 (2021)

  32. Bhosle, U.: Generalised parabolic bundles and applications to torsionfree sheaves on nodal curves. Ark. Mat. 30(2), 187–215 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  33. A. L. Muñoz-Castañeda, Generalized parabolic structures over smooth curves with many components and principal bundles over reducible nodal curves,http://arxiv.org/abs/1910.13403arXiv:1910.13403 (2020)

  34. W. Fulton, Young tableaux. With applications to representation theory and geometry, London Mathematical Society Student Texts. 35. Cambridge: Cambridge University Press, 1997

  35. Hartshorne, R.: Stable reflexive sheaves. Math. Ann. 254(2), 121–176 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  36. A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math (1965), no. 24

  37. W. Fulton, J. Harris Representation theory. A first course, Graduate Texts in Mathematics, 129. Readings in Mathematics. Springer-Verlag, New York, 1991

  38. Belkale, P., Kumar, S.: The multiplicative eigenvalue problem and deformed quantum cohomology. Adv. Math. 288, 1309–1359 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  39. Gómez, T.L., Langer, A., Schmitt, A.H.W., Sols, I.: Moduli spaces for principal bundles in arbitrary characteristic. Adv. Math. 219(4), 1177–1245 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  40. T. Gómez, I. Sols, Stable tensors and moduli space of orthogonal sheaves,arXiv: 0103150 (2001)

  41. Bhosle, U.: Tensor fields and singular principal bundles. Int. Math. Res. Not. 2004(57), 3057–3077 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  42. A. L. Muñoz-Castañeda, Principal G-bundles on nodal curves, Ph.D. thesis, Freie Universität Berlin (2017)

  43. C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. I, Inst. Hautes Études Sci. Publ. Math, (1994), no 79, 47-129

  44. Mumford, D., Fogarty, J., Kirwan, F.: Geometric invariant theory, Third, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34. Springer-Verlag, Berlin (1994)

    Google Scholar 

  45. Huybrechts, D., Lehn, M.: The geometry of moduli spaces of sheaves, 2nd edn. Cambridge Mathematical Library. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  46. Biswas, I., Hoffmann, N.: Poincaré families of G-bundles on a curve. Math. Ann. 352, 133–154 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  47. Ramanathan, A.: Stable principal bundles on a compact Riemann surface. Math. Ann. 213, 129–152 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  48. Holla, Y., Narasimhan, M.: A generalisation of Nagata’s theorem on ruled surfaces. Compositio Mathematica 127(3), 321–332 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  49. M.S. Narasimhan, T. R. Ramadas, Factorisation of generalised theta functions. I, Invent. Math. 114 (1993), no. 3, 565-623

  50. Kumar, S.: Conformal blocks, generalized theta functions and the Verlinde formula, new mathematical monographs. Cambridge University Press, Cambridge (2021)

    Book  Google Scholar 

  51. R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics. 52. New York-Heidelberg-Berlin: Springer-Verlag (1977)

  52. N. Fakhruddin, Chern classes of conformal blocks, Compact Moduli Spaces and Vector Bundles, Contemp. Math., vol. 564, Amer. Math. Soc., Providence, RI, 2012, pp. 145-176

  53. Drinfeld, V., Simpson, C.: \(B\)-structures on \(G\)-bundles and local triviality. Math. Res. Lett. 2(6), 823–829 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  54. Kumar, S.: Demazure character formula in arbitrary Kac-Moody setting. Invent Math 89, 395–423 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  55. P. Deligne, D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. (1969), no. 36, 75-109

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Acknowledgements

Thanks to P. Belkale, N. Fakhruddin, A. Gibney, S. Kumar, A. Muñoz-Castañeda, and A. Schmitt for useful comments and suggestions. Thank you also to the referees for many useful comments and corrections.

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  1. Department of Mathematics & Statistics, Kenyon College, Gambier, OH, 43022, USA

    Avery Wilson

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Wilson, A. Compactifications of Moduli of G-Bundles and Conformal Blocks. Transformation Groups (2023). https://doi.org/10.1007/s00031-023-09820-5

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