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**On the holomorphicity of genus two Lefschetz fibrations.**
*(English)*
Zbl 1090.53072

Much of the current research in smooth 4-manifold theory involves difficult questions that attempt to sort out the exact relationship between symplectic and holomorphic manifolds. This important paper proves three major theorems which can be viewed in this light. Theorem C demonstrates that any symplectic surface in \({\mathbb C}P^2\) of degree \(d\leq 17\) is symplectically isotopic to a complex algebraic curve. Theorem B is a similar result stating that symplectic surfaces in \(S^2\)-bundles over \(S^2\) of bidegree \((a,b)\) with \(a\leq 7\) are isotopic to holomorphic curves. Results concerning symplectic isotopy classes of symplectic curves have a long history, beginning with an influential paper of M. L. Gromov [Invent. Math. 82, 307–347 (1985; Zbl 0592.53025)], where he demonstrated Theorem C for curves of degree \(d\leq 2\). (Subsequent improvements were to \(d\leq 3\) by Sikorav, and \(d\leq 6\) by Shevchishin.) The conclusions of Theorems B and C are far from standard behavior, and a growing body of work is demonstrating that “symplectic knot theory” is in fact a rich subject; for instance, B. Ozbagci and A. Stipsicz [Proc. Am. Math. Soc. 128, 3125–3128 (2000; Zbl 0951.57015)] have shown that there is an infinite family of homologous non-isotopic symplectic surfaces in a blown up \(S^2\)-bundle over \(S^2\).

Theorem A is the result that gives the paper its title, that any genus two symplectic Lefschetz fibration with only irreducible fibers and transitive monodromy is holomorphic. A similar conclusion that all elliptic (genus one) symplectic Lefschetz fibrations are holomorphic has long been known, from work of Moishezon and LivnĂ© [in B. Moishezon, Complex surfaces and connected sums of complex planes, Lecture Notes in Math. 603, Springer-Verlag, New York (1977; Zbl 0392.32015)]. Their proof was essentially combinatorial, by finding standard forms for monodromy factorizations in the mapping class group \(SL(2,{\mathbb Z})\) of the torus. However the complexity of mapping class groups for surfaces of higher genus renders this approach unworkable, and necessitates the approach taken by Siebert and Tian. (In fact, nice corollaries of their paper are to give a purely geometric proof of the holomorphicity of elliptic fibrations, and to give standard braid monodromy factorizations for genus two fibrations.) Theorem A is proven as a straightforward consequence of Theorem B, by considering the fibration as a double cover of an \(S^2\)-bundle over \(S^2\) branched over a curve of bidegree \((6,b)\).

The heart of the paper is the proof of the isotopy lemma in section 8, from which Theorems B and C follow. This lemma is a technical statement regarding the uniqueness of isotopy classes of pseudo-holomorphic smoothings of a pseudo-holomorphic cycle \(C_\infty=\sum_a m_a C_{\infty,a}\) in an \(S^2\)-bundle over \(S^2\). This uniqueness holds under the assumption that \(c_1(M)\cdot C_{\infty,a}>0\) for all \(a\) (a natural assumption), and that

\[ \sum_{a| m_a>1} (c_1(M)\cdot C_{\infty,a} + g(C_{\infty,a})-1)<c_1(M)\cdot C_\infty-1. \]

This inequality is required to deal with problems that arise in deforming multiply covered components, and it is what leads to the degree bounds in Theorems B and C. To deal with non-multiple components, Siebert and Tian make clever use of a result in section 7, which allows them to replace non-multiple components by spheres while introducing only nodal singularities. Another key ingredient is work of Shevchishin on moduli spaces of pseudo-holomorphic curves [preprint, math.SG/0010262]. The full proof of the isotopy lemma is lengthy and impressive; a useful summary in the introduction alone runs for several pages.

Theorem A is the result that gives the paper its title, that any genus two symplectic Lefschetz fibration with only irreducible fibers and transitive monodromy is holomorphic. A similar conclusion that all elliptic (genus one) symplectic Lefschetz fibrations are holomorphic has long been known, from work of Moishezon and LivnĂ© [in B. Moishezon, Complex surfaces and connected sums of complex planes, Lecture Notes in Math. 603, Springer-Verlag, New York (1977; Zbl 0392.32015)]. Their proof was essentially combinatorial, by finding standard forms for monodromy factorizations in the mapping class group \(SL(2,{\mathbb Z})\) of the torus. However the complexity of mapping class groups for surfaces of higher genus renders this approach unworkable, and necessitates the approach taken by Siebert and Tian. (In fact, nice corollaries of their paper are to give a purely geometric proof of the holomorphicity of elliptic fibrations, and to give standard braid monodromy factorizations for genus two fibrations.) Theorem A is proven as a straightforward consequence of Theorem B, by considering the fibration as a double cover of an \(S^2\)-bundle over \(S^2\) branched over a curve of bidegree \((6,b)\).

The heart of the paper is the proof of the isotopy lemma in section 8, from which Theorems B and C follow. This lemma is a technical statement regarding the uniqueness of isotopy classes of pseudo-holomorphic smoothings of a pseudo-holomorphic cycle \(C_\infty=\sum_a m_a C_{\infty,a}\) in an \(S^2\)-bundle over \(S^2\). This uniqueness holds under the assumption that \(c_1(M)\cdot C_{\infty,a}>0\) for all \(a\) (a natural assumption), and that

\[ \sum_{a| m_a>1} (c_1(M)\cdot C_{\infty,a} + g(C_{\infty,a})-1)<c_1(M)\cdot C_\infty-1. \]

This inequality is required to deal with problems that arise in deforming multiply covered components, and it is what leads to the degree bounds in Theorems B and C. To deal with non-multiple components, Siebert and Tian make clever use of a result in section 7, which allows them to replace non-multiple components by spheres while introducing only nodal singularities. Another key ingredient is work of Shevchishin on moduli spaces of pseudo-holomorphic curves [preprint, math.SG/0010262]. The full proof of the isotopy lemma is lengthy and impressive; a useful summary in the introduction alone runs for several pages.

Reviewer: Terry Fuller (Northridge)