Geometry & Topology

Instantons on cylindrical manifolds and stable bundles

Brendan Owens

Full-text: Open access

Abstract

Let Σ be a smooth complex curve, and let S be the product ruled surface Σ×P1. We prove a correspondence conjectured by Donaldson between finite energy U(2)–instantons over Σ×S1×, and rank 2 holomorphic bundles over S whose restrictions to Σ×{0},Σ×{} are stable.

Article information

Source
Geom. Topol., Volume 5, Number 2 (2001), 761-797.

Dates
Received: 23 February 2001
Revised: 25 October 2001
Accepted: 5 October 2001
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513883043

Digital Object Identifier
doi:10.2140/gt.2001.5.761

Mathematical Reviews number (MathSciNet)
MR1871404

Zentralblatt MATH identifier
1069.53029

Subjects
Primary: 53C07: Special connections and metrics on vector bundles (Hermite-Einstein- Yang-Mills) [See also 32Q20]
Secondary: 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx] 57R58: Floer homology 14J80: Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants)

Keywords
Anti-self-dual connection stable bundle product ruled surface

Citation

Owens, Brendan. Instantons on cylindrical manifolds and stable bundles. Geom. Topol. 5 (2001), no. 2, 761--797. doi:10.2140/gt.2001.5.761. https://projecteuclid.org/euclid.gt/1513883043


Export citation

References

  • M Atiyah, R Bott The Yang–Mills equations over Riemann surfaces, Phil. Trans. R. Soc. Lond. A 308 (1982) 523–615
  • P Braam, S Donaldson, Floer's work on instanton homology, knots and surgery, from: “The Floer Memorial Volume”, Birkhäuser (1995) 195–281
  • B Booß-Bavnbek, K Wojciechowski, Elliptic boundary problems for Dirac operators, Birkhäuser (1993)
  • S Donaldson, Anti self-dual Yang–Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc. 50 (1985) 1–26
  • S Donaldson, Floer homology and algebraic geometry, from: “Vector Bundles in Algebraic Geometry: Durham 1993”, Cambridge University Press (1995)
  • S Donaldson, Boundary value problems for Yang–Mills fields, Journal of Geom. & Physics 8 (1992) 89–122
  • S Donaldson, P Kronheimer, The geometry of four-manifolds, Oxford University Press (1990)
  • S Dostoglou, D Salamon, Instanton homology and symplectic fixed points, from: “Symplectic Geometry”, London Math. Soc. Lecture Note Ser. 192, Cambridge Univ. Press (1993) 57–93
  • R Friedman, Algebraic surfaces and holomorphic vector bundles, Springer–Verlag (1998)
  • R Friedman, J Morgan, Smooth four-manifolds and complex surfaces, Springer–Verlag (1994)
  • D Gilbarg, N Trudinger, Elliptic partial differential equations of second order, second edition, Springer–Verlag (1983)
  • G-Y Guo, Yang–Mills fields on cylindrical manifolds and holomorphic bundles I, II, Comm. Math. Phys. 179 (1996) 737–775, 777–788
  • J Morgan, Comparison of the Donaldson polynomial invariants with their algebro-geometric analogues, Topology, 32 (1993) 449–488
  • J Morgan, T Mrowka, On the gluing theorem for instantons on manifolds containing long cylinders, preprint
  • J Morgan, T Mrowka, D Ruberman, The $L^2$ moduli space and a vanishing theorem for Donaldson polynomial invariants, International Press (1994)
  • P Newstead, Introduction to moduli problems and orbit spaces, Springer–Verlag (1978)
  • Z Qin, PhD Thesis, Columbia University (1990)