本项目主要研究近复流形与广义复流形的Kodaira维数和Hodge数。Kodaira维数和Hodge数是复流形上著名的不变量，Kodaira维数也可定义在四维辛流形上，为辛结构的双有理不变量。复流形和辛流形都容许近复结构，故一个自然问题是研究一般近复流形上的Kodaira维数和Hodge数。这将是建立近复流形双有理几何的关键步骤之一。我们首先研究近复流形上Kodaira维数的形变性和四维近复流形的双有理分类问题。其次，我们将研究近复流形上Hodge数对Hermitian度量的依赖性，对Hirzebruch问题二十作出部分解答。最后，我们研究Kodaira维数在广义复流形上的推广，希望通过将辛结构映射为广义复结构，定义出新的不变量。

The proposed project mainly investigates problems related to the Kodaira dimension and Hodge numbers on almost complex manifolds and generalized complex manifolds. Kodaira dimension and Hodge numbers are famous invariants on complex manifolds. Kodaira dimension can be also defined on 4-dimensional symplectic manifolds which turns out to be a symplectic birational invariant. Both complex and symplectic maniofolds admit almost complex structures. Therefore, the study of Kodaira dimension and Hodge number on almost complex manifolds is a natural generalization of the theory of complex and symplectic geometry. It would be a fundamental step to study the birational geometry of almost complex manifolds. First, we will study the deformation property of Kodaira dimension and birational classification of 4-dimensional almost complex manifolds. Next, we will study the harmonic Hodge number on almost complex manifolds, which is a part of the Hirzebruch problem 20. Finally, we will try to define the Kodaira dimension on generalized complex manifolds. By viewing symplectic structure as special generalized complex structures, we hope that we can obtain new invariant of symplectic structures.