Volume 6, issue 1 (2002)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Surface bundles over surfaces of small genus

Jim Bryan and Ron Donagi

Geometry & Topology 6 (2002) 59–67

arXiv: math.AG/0105203

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
Mathematical Subject Classification 2000
Primary: 14D05, 14D06, 57M20
Secondary: 57N05, 57N13, 14J29
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Publication
Received: 24 May 2001
Revised: 7 February 2002
Accepted: 26 February 2002
Published: 27 February 2002
Proposed: Dieter Kotschick
Seconded: Walter Neumann, Gang Tian
Authors
Jim Bryan
Department of Mathematics
University of British Columbia
121-1984 Mathematics Road
Vancouver
V6T 1Z2
British Columbia
Canada
Ron Donagi