On the holomorphicity of genus two Lefschetz fibrations

Bernd Siebert; Gang Tian

doi:10.4007/annals.2005.161.959

On the holomorphicity of genus two Lefschetz fibrations

Bernd Siebert, Gang Tian

Mathematics

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43 Scopus citations

Abstract

We prove that any genus-2 Lefschetz fibration without reducible fibers and with "transitive monodromy" is holomorphic. The latter condition comprises all cases where the number of singular fibers μ, ∈ 10ℕ is not congruent to 0 modulo 40. This proves a conjecture of the authors in [SiTil]. An auxiliary statement of independent interest is the holomorphicity of symplectic surfaces in S²-bundles over S², of relative degree ≤ 7 over the base, and of symplectic surfaces in ℂℙ ² of degree ≤ 17.

Original language	English (US)
Pages (from-to)	959-1020
Number of pages	62
Journal	Annals of Mathematics
Volume	161
Issue number	2
DOIs	https://doi.org/10.4007/annals.2005.161.959
State	Published - Mar 2005

All Science Journal Classification (ASJC) codes

Statistics and Probability
Statistics, Probability and Uncertainty

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AB - We prove that any genus-2 Lefschetz fibration without reducible fibers and with "transitive monodromy" is holomorphic. The latter condition comprises all cases where the number of singular fibers μ, ∈ 10ℕ is not congruent to 0 modulo 40. This proves a conjecture of the authors in [SiTil]. An auxiliary statement of independent interest is the holomorphicity of symplectic surfaces in S2-bundles over S2, of relative degree ≤ 7 over the base, and of symplectic surfaces in ℂℙ 2 of degree ≤ 17.

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On the holomorphicity of genus two Lefschetz fibrations

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