Rigidity of symplectic fillings, symplectic divisors and Dehn twist exact sequences

Abstract

We present three different aspects of symplectic geometry in connection to complex geometry. Convex symplectic manifolds, symplectic divisors and Lagrangians are central objects to study on the symplectic side. The focus of the thesis is to establish relations of these symplectic objects to the corresponding complex analytic objects, namely Stein fillings, divisors and coherent sheaves, respectively. Using pseudoholomorphic curve techniques and Gauge theoretic results, we system- atically study obstructions to symplectic/Stein fillings of contact 3-manifolds arising from the rigidity of closed symplectic four-manifolds with non-positive Kodaira dimen- sion. This perspective provides surprising consequences which, in particular, captures a new rigidity phenomenon for exact fillings of unit cotangent bundle of orientable surfaces and recovers many known results in a uniform way. The most important source of Stein fillings comes from smoothing of a complex isolated singularities. This motivates us to study when a symplectic divisor admits a convex/concave neighborhood and we obtain a complete and very computable answer to this local behaviour of symplectic divisors. Globally speaking, symplectic divisors in a closed symplectic manifold that represent its first Chern class are of particular importance in mirror symmetry. Such a symplectic divisor, together with the closed symplectic manifold together is called a symplectic log Calabi-Yau surface. We obtain a complete classification of symplectic log Calabi-Yau surface up to isotopy of symplectic divisors. Finally, we study algebraic properties of Fukaya category on the functor level and uti- lize Biran-Cornea’s Lagrangian cobordism theory and Mau-Wehrheim-Woodward func- tor to provide a partial proof of Huybrechts-Thomas’s conjecture.