This text is an expanded version of the lectures delivered by the authors at the CIME summer school “Symplectic 4-manifolds and algebraic surfaces,” Cetraro (Italy), September 2–10, 2003. The aim of these lectures were mostly to introduce graduate students to pseudo-holomorphic techniques for the study of isotopy of symplectic submanifolds in dimension four. We tried to keep the style of the lectures by emphasizing the basic reasons for the correctness of a result rather than by providing full details.
Essentially none of the content claims any originality, but most of the results are scattered in the literature in sometimes hard-to-read locations. For example, we give a hands-on proof of the smooth parametrization of the space of holomorphic cycles on a complex surface under some positivity assumption. This is usually derived by the big machinery of deformation theory together with Banach-analytic methods. For an uninitiated person it is hard not only to follow the formal arguments needed to deduce the result from places in the literature, but also, and maybe more importantly, to understand whyit is true. While our treatment here has the disadvantage to consider only a particular situation that does not even quite suffice for the proof of the Main Theorem (Theorem 9.1) we hope that it is useful for enhancing the understanding of this result outside the community of hardcore complex analysts and algebraic geometers.
The last section discusses the proof of the main theorem, which is also the main result from [SiTi3]. The logic of this proof is a bit difficult, involving several reduction steps and two inductions, and we are not sure if the current presentation really helps in understanding what is going on. Maybe somebody else has to take this up again and add new ideas into streamlining this piece.
Finally there is one section on the application to symplectic Lefschetz fibrations. This makes the link to the other lectures of the summer school, notably to those by Auroux and Smith.
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Siebert, B., Tian, G. (2008). Lectures on Pseudo-Holomorphic Curves and the Symplectic Isotopy Problem. In: Catanese, F., Tian, G. (eds) Symplectic 4-Manifolds and Algebraic Surfaces. Lecture Notes in Mathematics, vol 1938. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78279-7_5
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