Summary: Proceedings of 9th
pp, 1 10
Fiber sums of genus 2 Lefschetz fibrations
Abstract. Using the recent results of Siebert and Tian about the holomorphicity of
genus 2 Lefschetz fibrations with irreducible singular fibers, we show that any genus 2
Lefschetz fibration becomes holomorphic after fiber sum with a holomorphic fibration.
Symplectic Lefschetz fibrations have been the focus of a lot of attention since the proof
by Donaldson that, after blow-ups, every compact symplectic manifold admits such struc-
tures . Genus 2 Lefschetz fibrations, where the first non-trivial topological phenomena
arise, have been particularly studied. Most importantly, it has recently been shown by
Siebert and Tian that every genus 2 Lefschetz fibration without reducible fibers and with
"transitive monodromy" is holomorphic . The statement becomes false if reducible
singular fibers are allowed, as evidenced by the construction by Ozbagci and Stipsicz 
of genus 2 Lefschetz fibrations with non-complex total space (similar examples have also
been constructed by Ivan Smith; the reader is also referred to the work of Amorīos et al 
and Korkmaz  for related constructions).