Geometry & Topology

Surface bundles over surfaces of small genus

Jim Bryan and Ron Donagi

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We construct examples of non-isotrivial algebraic families of smooth complex projective curves over a curve of genus 2. This solves a problem from Kirby’s list of problems in low-dimensional topology. Namely, we show that 2 is the smallest possible base genus that can occur in a 4–manifold of non-zero signature which is an oriented fiber bundle over a Riemann surface. A refined version of the problem asks for the minimal base genus for fixed signature and fiber genus. Our constructions also provide new (asymptotic) upper bounds for these numbers.

Article information

Geom. Topol., Volume 6, Number 1 (2002), 59-67.

Received: 24 May 2001
Revised: 7 February 2002
Accepted: 26 February 2002
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14D05: Structure of families (Picard-Lefschetz, monodromy, etc.) 14D06: Fibrations, degenerations 57M20: Two-dimensional complexes
Secondary: 57N05: Topology of $E^2$ , 2-manifolds 57N13: Topology of $E^4$ , $4$-manifolds [See also 14Jxx, 32Jxx] 14J29: Surfaces of general type

Surface bundles 4–manifolds algebraic surface


Bryan, Jim; Donagi, Ron. Surface bundles over surfaces of small genus. Geom. Topol. 6 (2002), no. 1, 59--67. doi:10.2140/gt.2002.6.59.

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  • M F Atiyah, The signature of fibre-bundles, from: “Global Analysis (Papers in Honor of K. Kodaira)”, Univ. Tokyo Press, Tokyo (1969) 73–84
  • Jim Bryan, Ron Donagi, Andras Stipsicz, Surface bundles: some interesting examples, Turkish J. Math. 25 (2001) 61–68, proceedings of the $7^{th}$ Gökova Geometry and Topology conference
  • H Endo, M Korkmaz, D Kotschick, B Ozbagci, A Stipsicz, Commutators, Lefschetz fibrations and the signatures of surface bundles., to appear in Topology
  • Hisaaki Endo, A construction of surface bundles over surfaces with non-zero signature, Osaka J. Math. 35 (1998) 915–930
  • Gabino González-Díez, William J Harvey, On complete curves in moduli space. I, II, Math. Proc. Cambridge Philos. Soc. 110 (1991) 461–466, 467–472
  • Joe Harris, Ian Morrison, Moduli of curves, Springer–Verlag, New York (1998)
  • F Hirzebruch, The signature of ramified coverings, from: “Global Analysis (Papers in Honor of K. Kodaira)”, Univ. Tokyo Press, Tokyo (1969) 253–265
  • Rob Kirby, Problems in low dimensional topology, from: “Proceedings of the 1993 Georgia International Topology Conference held at the University of Georgia, Athens, GA, August 2–13, 1993”, (William H Kazez, editor), American Mathematical Society, Providence, RI (1997)
  • K Kodaira, A certain type of irregular algebraic surfaces, J. Analyse Math. 19 (1967) 207–215
  • D Kotschick, Signatures, monopoles and mapping class groups, Math. Res. Lett. 5 (1998) 227–234
  • Claude LeBrun, Diffeomorphisms, symplectic forms, and Kodaira fibrations, Geom. Topol. 4 (2000) 451–456
  • Ivan Smith, Lefschetz fibrations and the Hodge bundle, Geom. Topol. 3 (1999) 211–233