Stable pairs, linear systems and the Verlinde formula.

*(English)*Zbl 0882.14003The present paper, published more than three years ago, is to be seen as one of the milestones in the development of the moduli theory of vector bundles on algebraic curves and its applications to conformal quantum field theories. The main result is a general dimension formula for spaces of sections of line bundles on certain moduli spaces, which provides, as a special case, the celebrated Verlinde formula for spaces of generalized theta functions on the moduli spaces of semistable rank-2 vector bundles with fixed determinant on a smooth projective curve. Back then, in the early 1990’s, Thaddeus’s proof of the Verlinde formula was among the very first mathematically rigorous, affirmative establishments of this (formerly conjectural) dimension formula. In the meantime, as it is well-known, several other approaches to (and proofs of) the Verlinde formula have been presented, and that by various authors and in varying degrees of generality.

However, Thaddeus’s paper was and remains noteworthy, just because of the original, ingenious and far-reaching method that the author has developed in order to obtain his main result. Namely, his basic idea of tackling the space \(H^0(N, {\mathcal O} (k\cdot \Theta))\) of sections of \(k\)-th order theta functions, on the moduli space \(N\) of semi-stable rank-2 bundles and fixed determinant bundle \(L\) over a smooth projective curve \(X\), is to relate this space to the space of sections of another line bundle, not defined over \(N\), but over some derived, more manageable moduli space. To this end, the author studies so-called stable pairs of rank-2 bundles with fixed determinant and fixed non-trivial section of such a vector bundle.

There are several possible concepts of stability for such pairs, out of which the author uses the one that was introduced by S. Bradlow in 1991 [cf. S. B. Bradlow, J. Differ. Geom. 33, No. 1, 169-213 (1991; Zbl 0697.32014)]. This definition of stability for pairs \((E, \varphi)\), \(\varphi \in H^0 (X,E)\), depends on the choice of a real parameter \(\sigma\), and a good part of the author’s investigations is devoted to both establishing the existence of moduli spaces for such stable pairs, which are denoted by \(M (\sigma, L)\), and studying their dependence on the stability parameter \(\sigma\). This is inspired and modeled after D. Gieseker’s construction of the moduli spaces of stable vector bundles via geometric invariant theory. A very subtle analysis of the \(\sigma\)-stability condition for pairs \((E, \varphi)\) then leads to a tree of moduli spaces, partially related to each other by birational morphisms (“flips” in the sense of Mori), which finally relates the original moduli space \(N\) of semistable rank-2 vector bundles with determinant \(L\) to the “initial” moduli space \(M_0\) of the whole tree of moduli spaces. It turns out that the Verlinde space \(H^0 (N, (k\cdot \Theta))\) can be identified with the space of sections of an appropriate line bundle over any of the moduli spaces occurring in the tree, and that, at least for one of these moduli spaces, the higher cohomology of its “appropriate” line bundle vanishes. Thus the Verlinde number \(h^0(N, (k\cdot \Theta))\) can be computed as the Euler characteristic of some other moduli space, which, in turn, appears as the Euler characteristic of a line bundle over a symmetric product of the base curve \(X\).

Apart from establishing the Verlinde formula, in this way, the author obtains a whole family of dimension formulas for spaces of sections of line bundles over various moduli spaces, which might be important and useful for further investigations in related contexts. In addition, the rigorous establishing of moduli spaces for \(\sigma\)-stable pairs of vector bundles over curves represents a highly valuable contribution towards the moduli theory of varieties and vector bundles in its full generality.

However, Thaddeus’s paper was and remains noteworthy, just because of the original, ingenious and far-reaching method that the author has developed in order to obtain his main result. Namely, his basic idea of tackling the space \(H^0(N, {\mathcal O} (k\cdot \Theta))\) of sections of \(k\)-th order theta functions, on the moduli space \(N\) of semi-stable rank-2 bundles and fixed determinant bundle \(L\) over a smooth projective curve \(X\), is to relate this space to the space of sections of another line bundle, not defined over \(N\), but over some derived, more manageable moduli space. To this end, the author studies so-called stable pairs of rank-2 bundles with fixed determinant and fixed non-trivial section of such a vector bundle.

There are several possible concepts of stability for such pairs, out of which the author uses the one that was introduced by S. Bradlow in 1991 [cf. S. B. Bradlow, J. Differ. Geom. 33, No. 1, 169-213 (1991; Zbl 0697.32014)]. This definition of stability for pairs \((E, \varphi)\), \(\varphi \in H^0 (X,E)\), depends on the choice of a real parameter \(\sigma\), and a good part of the author’s investigations is devoted to both establishing the existence of moduli spaces for such stable pairs, which are denoted by \(M (\sigma, L)\), and studying their dependence on the stability parameter \(\sigma\). This is inspired and modeled after D. Gieseker’s construction of the moduli spaces of stable vector bundles via geometric invariant theory. A very subtle analysis of the \(\sigma\)-stability condition for pairs \((E, \varphi)\) then leads to a tree of moduli spaces, partially related to each other by birational morphisms (“flips” in the sense of Mori), which finally relates the original moduli space \(N\) of semistable rank-2 vector bundles with determinant \(L\) to the “initial” moduli space \(M_0\) of the whole tree of moduli spaces. It turns out that the Verlinde space \(H^0 (N, (k\cdot \Theta))\) can be identified with the space of sections of an appropriate line bundle over any of the moduli spaces occurring in the tree, and that, at least for one of these moduli spaces, the higher cohomology of its “appropriate” line bundle vanishes. Thus the Verlinde number \(h^0(N, (k\cdot \Theta))\) can be computed as the Euler characteristic of some other moduli space, which, in turn, appears as the Euler characteristic of a line bundle over a symmetric product of the base curve \(X\).

Apart from establishing the Verlinde formula, in this way, the author obtains a whole family of dimension formulas for spaces of sections of line bundles over various moduli spaces, which might be important and useful for further investigations in related contexts. In addition, the rigorous establishing of moduli spaces for \(\sigma\)-stable pairs of vector bundles over curves represents a highly valuable contribution towards the moduli theory of varieties and vector bundles in its full generality.

Reviewer: W.Kleinert (Berlin)

##### MSC:

14D20 | Algebraic moduli problems, moduli of vector bundles |

14H60 | Vector bundles on curves and their moduli |

14C20 | Divisors, linear systems, invertible sheaves |

14F25 | Classical real and complex (co)homology in algebraic geometry |

32G08 | Deformations of fiber bundles |

##### Keywords:

flips; moduli theory of vector bundles; conformal quantum field theories; dimension formula; Verlinde formula; theta functions; Verlinde space##### References:

[1] | Arbarello, E., Cornalba, M., Griffiths, P.A., Harris, J.: Geometry of algebraic curves, vol. I. Berlin Heidelberg New York: Springer, 1985 · Zbl 0559.14017 |

[2] | Atiyah, M.F., Bott, R.: The Yang-Mills equations over Riemann surfaces. Philos. Trans. R. Soc. Lond., A308, 523-615 (1982) · Zbl 0509.14014 · doi:10.1098/rsta.1983.0017 |

[3] | Bertram, A.: Moduli of rank-2 vector bundles, theta divisors, and the geometry of curves in projective space. J. Differ. Geom.35, 429-469 (1992) · Zbl 0787.14014 |

[4] | Bertram, A.: Stable pairs and stable parabolic pairs. (Harvard Preprint) · Zbl 0826.14017 |

[5] | Bradlow, S.: Special metrics and stability for holomorphic bundles with global sections. J. Differ. Geom.33, 169-213 (1991) · Zbl 0697.32014 |

[6] | Bradlow, S., Daskalopoulos, G.: Moduli of stable pairs for holomorphic bundles over Riemann surfaces. Int. J. Math.2, 477-513 (1991) · Zbl 0759.32013 · doi:10.1142/S0129167X91000272 |

[7] | Dijkgraaf, R., Verlinde, E.: Modular invariance and the fusion algebra. Nucl. Phys. B (Proc. Suppl.)5B, 87-97 (1988) · Zbl 0958.81510 · doi:10.1016/0920-5632(88)90371-4 |

[8] | Donaldson, S.K.: Instantons in Yang-Mills theory. In: Quillen, D.G., Segal, G.B., Tsou, S.T. (eds.) The interface of mathematics and particle physics. Oxford, Oxford University Press 1990 |

[9] | Drezet, J.-M., Narasimhan, M.S.: Groupe de Picard des variétés de modules de fibrés semistables sur les courbes algébriques. Invent. Math.97, 53-94 (1989) · Zbl 0689.14012 · doi:10.1007/BF01850655 |

[10] | Garcia-Prada, O.: Dimensional reduction of stable bundles, vortices and stable pairs (in preparation) |

[11] | Gieseker, D.: On the moduli of vector bundles on an algebraic surface. Ann. Math.106, 45-60 (1977) · Zbl 0381.14003 · doi:10.2307/1971157 |

[12] | Griffiths, P., Harris, J.: Principles of algebraic geometry. New York: Wiley 1978 · Zbl 0408.14001 |

[13] | Grothendieck, A.: Technique de descente et théorèmes d’existence en géométrie algébrique, IV: Les schémas de Hilbert. Sémin. Bourbaki 1960-61, exp. 221; reprinted in: Fondements de la géométrie algébrique. Paris: Secrétariat Math. 1962 |

[14] | Harder, G., Narasimhan, M.S.: On the cohomology groups of moduli spaces of vector bundles over curves. Math. Ann.212, 215-248 (1975) · Zbl 0324.14006 · doi:10.1007/BF01357141 |

[15] | Hartshorne, R.: Algebraic geometry. Berlin Heidelberg New York: Springer 1977 · Zbl 0367.14001 |

[16] | Lazarsfeld, R.: A sampling of vector bundle techniques in the study of linear series. In: Cornalba, M., Gomez-Mont, X., Verjovsky, A. (eds.) Lectures on Riemann surfaces, pp. 500-559. World Scientific 1989 · Zbl 0800.14003 |

[17] | Macdonald, I.G.: Symmetric products of an algebraic curve. Topology1, 319-343 (1962) · Zbl 0121.38003 · doi:10.1016/0040-9383(62)90019-8 |

[18] | Mumford, D.: The red book of varieties and schemes. (Lect. Notes Math., Vol. 1358) Berlin Heidelberg New York: Springer 1988 · Zbl 0658.14001 |

[19] | Mumford, D., Fogarty, J.: Geometric invariant theory, second enlarged edition. Berlin Heidelberg New York: Springer 1982 · Zbl 0504.14008 |

[20] | Narasimhan, M.S., Ramanan, S.: Moduli of vector bundles on a compact Riemann surface. Ann. Math.89, 1201-1208 (1969) · Zbl 0186.54902 · doi:10.2307/1970807 |

[21] | Newstead, P.E.: Introduction to moduli problems and orbit spaces. Bombay: Tata Inst. 1978 · Zbl 0411.14003 |

[22] | Okonek, C., Schneider, M., Spindler, H., Vector bundles on complex projective spaces. Boston Basel Stuttgart: Birkhäuser 1980 · Zbl 0438.32016 |

[23] | Thaddeus, M.: A finite-dimensional approach to Verlinde’s factorization principle. (Preprint) |

[24] | Verlinde, E.: Fusion rules and modular transformations in 2d conformal field theory. Nucl. Phys.B 300, 360-376 (1988) · Zbl 1180.81120 · doi:10.1016/0550-3213(88)90603-7 |

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