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Torelli theorem for the moduli spaces of pairs


Muñoz, Vicente (2009) Torelli theorem for the moduli spaces of pairs. Mathematical Proceedings of the Cambridge Philosophical Society, 146 (3). pp. 675-693. ISSN 0305-0041

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Let X be a smooth projective curve of genus g >= 2 over C. A pair (E, phi) over X consists of an algebraic vector bundle E over X and a section phi is an element of H(0)(E). There is a concept of stability for pairs which depends on a real parameter tau. Here we prove that the third cohomology groups of the moduli spaces of tau-stable pairs with fixed determinant and rank n >= 2 are polarised pure Hodge structures, and they are isomorphic to H(1) (X) with its natural polarisation (except in very few exceptional cases). This implies a Torelli theorem for such moduli spaces. We recover that the third cohomology group of the moduli space of stable bundles of rank n >= 2 and fixed determinant is a polarised pure Hodge structure, which is isomorphic to H(1) (X). We also prove Torelli theorems for the corresponding moduli spaces of pairs and bundles with non-fixed determinant.

Item Type:Article
Uncontrolled Keywords:Ppolystable pair; Semistable vector bundles; Semistable triple; Moduli space; Smooth projective curve; Torelli theorem; Hodge structure
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:17030

D. ARAPURA and P. SASTRY. Intermediate Jacobians and Hodge structures of moduli spaces. Proc.Indian Acad. Sci. Math. Sci. 110 (2000), 1–26.

V.BALAJI and P. A. VISHWANATH. Deformations of Picard sheaves and moduli of pairs. Duke Math.J. 76 (1994), no. 3, 773–792.

A. BERTRAM. Stable pairs and stable parabolic pairs. J. Algebraic Geom. 3 (1994), no. 4, 703–724.

I. BISWAS and V. MUÑOZ. The Torelli theorem for the moduli spaces of connections on a Riemann surface. Topology 46 (2007), no. 3, 295–317.

S. B. BRADLOW and DASKALOPOULOS. Moduli of stable pairs for holomorphic bundles over Riemann surfaces. Internat. J. Math. 2 (1991), 477–513.

S. B. BRADLOW and O. GARC´IA–PRADA. Stable triples, equivariant bundles and dimensional reduction.Math. Ann. 304 (1996), 225–252.

S. B. BRADLOW, O. GARCIA–PRADA and P. B. GOTHEN. Moduli spaces of holomorphic triples over compact Riemann surfaces. Math. Ann. 328 (2004), 299–351.

S. B. BRADLOW, O. GARCIA-PRADA, V. MU ˜NOZ and P. E. NEWSTEAD. Coherent systems and Brill-Noether theory. Internat. J. Math. 14 (2003), no. 7, 683–733.

P. DELIGNE. Theorie de Hodge I,II,III. In Proc. I.C.M., vol. 1, 1970, pp. 425–430; in Publ. Math.I.H.E.S. 40 (1971), 5–58; ibid. 44 (1974), 5–77.

O. GARCIA–PRADA. Dimensional reduction of stable bundles, vortices and stable pairs. Internat. J.Math. 5 (1994), 1–52.

A.KOUVIDAKIS and T. PANTEV. The automorphism group of the moduli space of semistable vector bundles. Math. Ann. 302 (1995), 225–268.

D.MUMFORD and P. NEWSTEAD. Periods of a moduli space of bundles on curves. Amer. J. Math. 90 (1968), 1200–1208.

V. MUÑOZ. Hodge polynomials of the moduli spaces of rank 3 pairs. Geometriae Dedicata. 136 (2008), 17–46.

V.MUÑOZ, D. ORTEGA and M-J. VÁZQUEZ-GALLO. Hodge polynomials of the moduli spaces of pairs. Internat. J. Math. 18 (2007), 695–721.

M. S. NARASIMHAN and S. RAMANAN. Geometry of Hecke cycles. I. C. P. Ramanujam—a tribute,pp. 291–345, Tata Inst. Fund. Res. Studies in Math. 8 (Springer, 1978).

M. S. NARASIMHAN and S. RAMANAN. Moduli of vector bundles on a compact Riemann surface,Ann. of Math. (2) 89 (1969),14–51.

A. SCHMITT. A universal construction for the moduli spaces of decorated vector bundles. Transform. Groups 9 (2004), 167–209.

M.THADDEUS. Stable pairs, linear systems and the Verlinde formula. Invent. Math. 117 (1994), 317–353.

A. N. TYURIN. Geometry of moduli of vector bundles. Russ. Math. Surverys 29 (1974), 59–88.

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