Duke Mathematical Journal

Virtual intersections on the Quot scheme and Vafa-Intriligator formulas

Alina Marian and Dragos Oprea

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Abstract

We construct a virtual fundamental class on the Quot scheme parametrizing quotients of a trivial bundle on a smooth projective curve. We use the virtual localization formula to calculate virtual intersection numbers on Quot. As a consequence, we re-prove the Vafa-Intriligator formula; our answer is valid even when the Quot scheme is badly behaved. More intersections of Vafa-Intriligator type are computed by the same method. Finally, we present an application to the nonvanishing of the Pontryagin ring of the moduli space of bundles

Article information

Source
Duke Math. J., Volume 136, Number 1 (2007), 81-113.

Dates
First available in Project Euclid: 4 December 2006

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1165244880

Digital Object Identifier
doi:10.1215/S0012-7094-07-13613-5

Mathematical Reviews number (MathSciNet)
MR2271296

Zentralblatt MATH identifier
1117.14055

Subjects
Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
Secondary: 14C17: Intersection theory, characteristic classes, intersection multiplicities [See also 13H15] 14H60: Vector bundles on curves and their moduli [See also 14D20, 14F05]

Citation

Marian, Alina; Oprea, Dragos. Virtual intersections on the $\mathrm{Quot}$ scheme and Vafa-Intriligator formulas. Duke Math. J. 136 (2007), no. 1, 81--113. doi:10.1215/S0012-7094-07-13613-5. https://projecteuclid.org/euclid.dmj/1165244880


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References

  • E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of Algebraic Curves, Vol. 1, Grundlehren Math. Wiss. 267, Springer, New York, 1985.
  • K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), 45--88.
  • A. Bertram, Towards a Schubert calculus for maps from a Riemann surface to a Grassmannian, Internat. J. Math. 5 (1994), 811--825.
  • A. Bertram, G. Daskalopoulos, and R. Wentworth, Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians, J. Amer. Math. Soc. 9 (1996), 529--571.
  • R.-O. Buchweitz and H. Flenner, A semiregularity map for modules and applications to deformations, Compositio Math. 137 (2003), 135--210.
  • I. Ciocan-Fontanine and M. Kapranov, Derived Quot schemes, Ann. Sci. École Norm. Sup. (4) 34 (2001), 403--440.
  • —, Virtual fundamental classes for dg-manifolds, unpublished manuscript.
  • M. Duerr [DüRr], A. Kabanov, and C. Okonek, Poincaré invariants, preprint.
  • R. Earl and F. Kirwan, The Pontryagin rings of moduli spaces of arbitrary rank holomorphic bundles over a Riemann surface, J. London Math. Soc. (2) 60 (1999), 835--846.
  • T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), 487--518.
  • A. Grothendieck, ``Techniques de construction et théorèmes d'existence en géométrie algébrique, IV: Les schémas de Hilbert'' in Séminaire Bourbaki, Vol. 6 (1960/1961), exp. no. 221, Soc. Math. France, Montrouge, 1995, 249--276.
  • K. Intriligator, Fusion residues, Modern Phys. Lett. A 6 (1991), 3543--3556.
  • L. C. Jeffrey and F. C. Kirwan, Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface, Ann. of Math. (2) 148 (1998), 109--196.
  • B. Kim, A. Kresch, and T. Pantev, Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee, J. Pure Appl. Algebra 179 (2003), 127--136.
  • J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998), 119--174.
  • A. Marian, Intersection theory on the moduli space of stable bundles via morphism spaces, Ph.D. dissertation, Harvard University, Cambridge, Mass., 2004.
  • A. Marian and D. Oprea, On the intersection theory of the moduli space of rank two bundles, Topology 45 (2006), 531--541.
  • P. E. Newstead, Characteristic classes of stable bundles of rank $2$ over an algebraic curve, Trans. Amer. Math. Soc. 169 (1972), 337--345.
  • B. Siebert, ``Virtual fundamental classes, global normal cones and Fulton's canonical classes'' in Frobenius Manifolds, Aspects Math. E36, Vieweg, Wiesbaden, Germany, 2004, 341--358.
  • B. Siebert and G. Tian, On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator, Asian J. Math. 1 (1997), 679--695.
  • S. A. StrøMme, ``On parametrized rational curves in Grassmann varieties'' in Space Curves (Rocca di Papa, Italy, 1958), Lecture Notes in Math. 1266, Springer, Berlin, 1987, 251--272.
  • M. Thaddeus, Conformal field theory and the cohomology of the moduli space of stable bundles, J. Differential Geom. 35 (1992), 131--149.
  • R. P. Thomas, A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on $K3$ fibrations, J. Differential Geom. 54 (2000), 367--438.