Geodesic
On the holomorphicity of genus two Lefschetz fibrations
Bernd Siebert1 ; Gang Tian2
1 Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, 79104 Freiburg, Germany
2 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, United States
Annals of mathematics, Tome 161 (2005) no. 2, pp. 959-1020.

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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 $\mu\in 10\mathbb{N}$ is not congruent to $0$ modulo $40$. This proves a conjecture of the authors in [SiTi1]. An auxiliary statement of independent interest is the holomorphicity of symplectic surfaces in $S^2$-bundles over $S^2$, of relative degree $\le 7$ over the base, and of symplectic surfaces in $\mathbb{CP}^2$ of degree $\le 17$.
DOI : 10.4007/annals.2005.161.959

Bernd Siebert 1 ; Gang Tian 2

1 Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, 79104 Freiburg, Germany
2 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, United States 
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Bernd Siebert; Gang Tian. On the holomorphicity of genus two Lefschetz fibrations. Annals of mathematics, Tome 161 (2005) no. 2, pp. 959-1020. doi : 10.4007/annals.2005.161.959. http://geodesic.mathdoc.fr/articles/10.4007/annals.2005.161.959/

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