Symplectic geometry and its connection with complex geometry.

摘要

In this thesis, we will study the topology of symplectic manifolds and emphasis on its connection with the properties of complex manifolds.;We generalize the concept of second order real Dolbeault cohomololgy groups to almost complex structures. We prove it is indeed a direct sum decomposition in dimension 4. By this notion, we study the almost Kahler cone (or J-compatible cone) from various aspects. First, we obtain a comparison theorem on almost Kahler cone and J-tamed cone for any even dimension. In turn, we study the Nakai Moishezon type Theorem and Donaldson's "tamed to compatible" question for almost complex structures on rational surfaces. We also confirm Donaldson's question on complex surfaces.;Kodaira dimension is an important notion for complex manifolds. We confirm that the symplectic Kodaira dimension in dimension 4 is equal to the complex Kodaira dimension when the manifolds admit both structures. We also introduce 3-dimensional analogue and a relative version for symplectic manifolds in dimension 4 (and 2). By this notion, we confirm the Kodaira dimensions are additive in most interesting cases.