Variations of moduli of parabolic bundles.

*(English)*Zbl 0821.14007In this paper, we analyze the effect of varying the weights on the moduli of semistable parabolic bundles. It is observed that for generic weights, the moduli space is a smooth projective variety. Further, the “singular” weights form (real) hyperplanes. The moduli of parabolic bundles with weights \(\alpha\) and \(\beta \) on either side of one of these hyperplanes are related by a special type of birational transformation which is similar to a flip in Mori theory, i.e. a generalized “blow- down” followed by a “blow-up” along the singular locus of the singular moduli with weights \(\gamma\) on this hyperplane.

To prove this, first the singular locus is identified with the product of two moduli of parabolic bundles of lower rank. Then natural inclusion maps are proved to exist on the level of semistable holomorphic structures between the various types of stability (i.e. between \(\alpha\), \(\beta\) and \(\gamma\) stable and semistable bundles). These inclusions descend to the level of moduli to give the necessary birational maps, and it is demonstrated that these maps are isomorphisms off the singular locus. The result is complete by identifying the fiber over a point in the singular locus with the projective space on the nontrivial (parabolic) bundle extensions, and by showing that the dimensions of the two projective fibers (one for \(\alpha\) and the other for \(\beta)\) are in a precise sense complementary.

An immediate consequence of this is a formula for the intersection homology of a singular moduli in terms of the homology of either of the two smooth moduli (homology and intersection homology is taken with rational coefficients and middle perversity). For example, one of the smooth moduli is a small resolution of the singular moduli (in the sense of Goresky-MacPherson). Also, as a corollary, one gets a simple formula for the difference of the Poincaré polynomials of the two smooth moduli in terms of the Poincaré polynomial of the product of moduli of lower rank, i.e. the singular locus. Furthermore, this formula holds for integral homology, as a previous paper of the first author proves that the smooth moduli spaces are torsion free.

Further applications of the main theorem are given in the final sections. The case of very singular moduli (i.e. when the weights lie on the intersection of many hyperplanes) are considered and a natural stratification on these singular moduli is introduced. A stratified version of the main theorem is proved with respect to these natural stratifications. For example, when one of the weights is generic, the natural morphism between the moduli is a smooth resolution of the singular moduli. It is conjectured that for every singular moduli, there is a nearby smooth moduli so that the canonical morphism is a small resolution.

For ease of exposition in the earlier sections, we make the simplifying assumptions that there is only one parabolic point and the genus is larger than one. But these assumptions are not essential. In the final section, we outline how to adapt the results to the case of arbitrarily many parabolic points and any genus. Further, it is observed that the same results hold for moduli with fixed determinant. To conclude, rationality of the fixed determinant moduli is proved in the case of genus zero and discussed in the other cases. We close with a question as to the existence of an analogue to the Hecke correspondence in the setting of parabolic bundles.

To prove this, first the singular locus is identified with the product of two moduli of parabolic bundles of lower rank. Then natural inclusion maps are proved to exist on the level of semistable holomorphic structures between the various types of stability (i.e. between \(\alpha\), \(\beta\) and \(\gamma\) stable and semistable bundles). These inclusions descend to the level of moduli to give the necessary birational maps, and it is demonstrated that these maps are isomorphisms off the singular locus. The result is complete by identifying the fiber over a point in the singular locus with the projective space on the nontrivial (parabolic) bundle extensions, and by showing that the dimensions of the two projective fibers (one for \(\alpha\) and the other for \(\beta)\) are in a precise sense complementary.

An immediate consequence of this is a formula for the intersection homology of a singular moduli in terms of the homology of either of the two smooth moduli (homology and intersection homology is taken with rational coefficients and middle perversity). For example, one of the smooth moduli is a small resolution of the singular moduli (in the sense of Goresky-MacPherson). Also, as a corollary, one gets a simple formula for the difference of the Poincaré polynomials of the two smooth moduli in terms of the Poincaré polynomial of the product of moduli of lower rank, i.e. the singular locus. Furthermore, this formula holds for integral homology, as a previous paper of the first author proves that the smooth moduli spaces are torsion free.

Further applications of the main theorem are given in the final sections. The case of very singular moduli (i.e. when the weights lie on the intersection of many hyperplanes) are considered and a natural stratification on these singular moduli is introduced. A stratified version of the main theorem is proved with respect to these natural stratifications. For example, when one of the weights is generic, the natural morphism between the moduli is a smooth resolution of the singular moduli. It is conjectured that for every singular moduli, there is a nearby smooth moduli so that the canonical morphism is a small resolution.

For ease of exposition in the earlier sections, we make the simplifying assumptions that there is only one parabolic point and the genus is larger than one. But these assumptions are not essential. In the final section, we outline how to adapt the results to the case of arbitrarily many parabolic points and any genus. Further, it is observed that the same results hold for moduli with fixed determinant. To conclude, rationality of the fixed determinant moduli is proved in the case of genus zero and discussed in the other cases. We close with a question as to the existence of an analogue to the Hecke correspondence in the setting of parabolic bundles.

Reviewer: H.U.Boden und Y.Hu (Ann Arbor)

##### MSC:

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

14F43 | Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) |

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

##### Keywords:

weights; moduli of semistable parabolic bundles; birational transformation; flip; singular locus; intersection homology; moduli with fixed determinant##### References:

[1] | M. Atiyah and R. Bott, The Yang-Mills equations on a Riemann surface, Phil. Trans. Roy. Soc. Lond. A.,308, (1982), 524-615. · Zbl 0509.14014 |

[2] | S. Bauer, Parabolic bundles, elliptic surfaces and SU(2)-representation spaces of genus zero Fuchsian groups, Math. Ann.,290 (1991), 509-526. · Zbl 0752.14035 · doi:10.1007/BF01459257 |

[3] | A. Bertram and Q. Szenes, Hilbert Polynomials of moduli spaces of rank 2 vector bundles II, Topology,32 (1993) 599-609. · Zbl 0798.14004 · doi:10.1016/0040-9383(93)90011-J |

[4] | H.U. Boden, Representations of orbifold groups and parabolic bundles, Comm. Math. Helv.,66 (1991), 389-447. · Zbl 0758.57013 · doi:10.1007/BF02566657 |

[5] | H.U. Boden, Unitary Representations of Brieskorn spheres, MPI preprint 1993, to appear in Duke Math. J. |

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

[7] | G. Daskalopoulos and R. Wentworth, Geometric quantization for the moduli space of vector bundles with parabolic structure, preprint, 1991, to appear in Duke Math. Journal. · Zbl 0929.53049 |

[8] | I. Dolgachev and Y. Hu, Variation of Geometric Invariant Theory Quotients, preprint (1994), alg-geom/9402008. |

[9] | C. Frohman and A. Nicas, An intersection homology invariant for knots in a rational homology sphere, Topology,33 (1994) 123-158. · Zbl 0822.57008 · doi:10.1016/0040-9383(94)90039-6 |

[10] | W. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. Math.,54 (1984), 200-225. · Zbl 0574.32032 · doi:10.1016/0001-8708(84)90040-9 |

[11] | R.M. Goresky, R.D. MacPherson, On the topology of algebraic torus actions, Lecture Notes in Math. 1271, 73-90, Springer-Verlag 1986. · Zbl 0633.14025 |

[12] | R.M. Goresky and R.D. MacPherson, Intersection Homology II., Inven. Math.,71 (1983), 77-129. · Zbl 0529.55007 · doi:10.1007/BF01389130 |

[13] | A. Grothendieck, Sur la classification des fibrés holomorphes sur la sphère de Riemann, Amer. J. Math.,79 (1957), 121-138. · Zbl 0079.17001 · doi:10.2307/2372388 |

[14] | A. Grothendieck and J. Dieudonné, Éléments de géométrie algeéebrique III, (Seconde Partie). Inst. Hautes Études Sci. Publ. Math,17 (1963). |

[15] | V. Guillemin and S. Sternberg, Birational equivalence in symplectic category, Invent. Math.,97 (1989), 485-522. · Zbl 0683.53033 · doi:10.1007/BF01388888 |

[16] | F. Hirzebruch, Topological Methods in Algebraic Geometry, Springer-Verlag, New York 1978. · Zbl 0376.14001 |

[17] | Y. Hu, The geometry and topology of quotient varieties of torus actions, Duke Math. Journal,68, (1992), 151-184. · Zbl 0812.14031 · doi:10.1215/S0012-7094-92-06806-2 |

[18] | F. Kirwan, On the homology of compactifications of moduli spaces of vector bundles over a Riemann surface. Proc. Lond. Math. Soc.,53 (1986), 237-266. · Zbl 0607.14017 · doi:10.1112/plms/s3-53.2.237 |

[19] | V. Mehta and C. Seshadri, Moduli of vector bundles on smooth curves with parabolic structures, Math. Ann.,248 (1980), 205-239. · Zbl 0454.14006 · doi:10.1007/BF01420526 |

[20] | D. Mumford and J. Fogarty, Geometric Invariant Theory, A Series of Modern Surveys in Mathematics, Springer-Verlag (1982). · Zbl 0504.14008 |

[21] | P. Newstead, Topological properties of some spaces of stable bundles, Topology,6 (1967) 241-262. · Zbl 0201.23401 · doi:10.1016/0040-9383(67)90037-7 |

[22] | P. Newstead, Rationality of moduli spaces of stable bundles, Math. Ann.,215, (1975) 251-268, Math. Ann.,249 (1980), 281-282. · Zbl 0295.14004 · doi:10.1007/BF01343893 |

[23] | N. Nitsure, Cohomology of the moduli of parabolic bundles, Proc. Ind. Math. Soc.,95 (1986), 61-77. · Zbl 0611.14014 · doi:10.1007/BF02837250 |

[24] | C. Seshadri, Fibrés vectoriels sur les courbes algébriques, Asterisque,96, (1982). |

[25] | S. Smale, The structure of manifolds, Amer. J. Math.,84 (1962), 387-399. · Zbl 0109.41103 · doi:10.2307/2372978 |

[26] | M. Thaddeus, Stable pairs, linear systems and the Verlinde formula, Inven. Math.,117 (1994), 317-353. · Zbl 0882.14003 · doi:10.1007/BF01232244 |

[27] | L. Tu, Semistable bundles over an elliptic curve. Adv. in Math.,98, 1-26 (1993). · Zbl 0786.14021 · doi:10.1006/aima.1993.1011 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.