TAILIEUCHUNG - Đề tài " On the holomorphicity of genus two Lefschetz fibrations "

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 [SiTi1]. An auxiliary statement of independent interest is the holomorphicity of symplectic surfaces in S 2 -bundles over S 2 , of relative degree ≤ 7 over the base, and of symplectic surfaces in CP2 of degree ≤ 17. . | Annals of Mathematics On the holomorphicity of genus two Lefschetz fibrations By Bernd Siebertn and Gang Tian Annals of Mathematics 161 2005 959 1020 On the holomorphicity of genus two Lefschetz fibrations By Bernd Siebert and Gang Tian 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 p E 10N 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 S2-bundles over S2 of relative degree 7 over the base and of symplectic surfaces in CP2 of degree 17. Contents Introduction 1. Pseudo-holomorphic S2-bundles 2. Pseudo-holomorphic cycles on pseudo-holomorphic S2-bundles 3. The C0-topology on the space of pseudo-holomorphic cycles 4. Unobstructed deformations of pseudo-holomorphic cycle 5. Good almost complex structures 6. Generic paths and smoothings 7. Pseudo-holomorphic spheres with prescribed singularities 8. An isotopy lemma 9. Proofs of Theorems A B and C References Introduction A differentiable Lefschetz fibration of a closed oriented four-manifold M is a differentiable surjection p M - S2 with only finitely many critical points of the form t o p z w zw. Here z w and t are complex coordinates on M and S2 respectively that are compatible with the orientations. This generalization of classical Lefschetz fibrations in Algebraic Geometry was introduced Supported by the Heisenberg program of the DFG. Supported by NSF grants and a J. Simons fund. 960 BERND SIEBERT AND GANG TIAN by Moishezon in the late seventies for the study of complex surfaces from the differentiable viewpoint Mol . It is then natural to ask how far differentiable Lefschetz fibrations are from holomorphic ones. This question becomes even more interesting in view of Donaldson s result on the existence of symplectic Lefschetz pencils

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