Geometry & Topology, Vol. 6 (2002) Paper no. 3, pages 59--67.

Surface bundles over surfaces of small genus

Jim Bryan, Ron Donagi

Abstract. 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.

Keywords. Surface bundles, 4-manifolds, algebraic surface

AMS subject classification. Primary: 14D05, 14D06, 57M20. Secondary: 57N05, 57N13, 14J29.

DOI: 10.2140/gt.2002.6.59

E-print: arXiv:math.AG/0105203

Submitted to GT on 24 May 2001. (Revised 7 February 2002.) Paper accepted 26 February 2002. Paper published 27 February 2002.

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Jim Bryan, Ron Donagi
Department of Mathematics, University of British Columbia
121-1984 Mathematics Road, Vancouver BC
Canada V6T 1Z2
and
Department of Mathematics, University of Pennsylvania
209 S 33rd Street, Philadelphia, PA 19104-6395, USA

Email: jbryan@math.ubc.ca, donagi@math.upenn.edu

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