#### 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.

Notes on file formats
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

GT home page

## Archival Version

**These pages are not updated anymore.
They reflect the state of
.
For the current production of this journal, please refer to
http://msp.warwick.ac.uk/.
**