我们计划进一步发展L.Hormander, J.Kohn, C.B.Morrey等人在多复变函数论中建立起来的关于经典柯西－黎曼算子的L2方法，并用来研究复流形/辛流形中的整/伪全纯曲线。建立伪全纯曲线横截性的判据是关于伪全纯曲线形变的基本问题之一。著名辛拓扑学家D.Mcduff和D.Salamon在多个文献、专著中明确指出：在高维近复流形中，现在还没有一个有效的方法建立关于伪全纯曲线自横截性的判据。伪全纯曲线的形变问题的研究可以归结为研究一类实Cauchy－Riemann方程组(一般不是通常的复Cauchy－Riemann方程组)，我们计划对这类方程组建立Hormander－Kohn－Morrey理论并进一步研究辛流形中伪全纯曲线的自动横截性与超刚性猜想, 以及复流形中整全纯曲线(几何双曲问题)。

This project aims to develop the L2 theory due to L.Hormander, J.Kohn and C.B.Morrey which is established for the d-bar operator in SCV, and use it to attack important problems concerning entire/pseudo holomorphic curves in complex and symplectic geometry. One of the fundamental questions in the deformation theory of pseudoholomorphic curves is to find criteria for transversality. As pointed out by Professors Mucduff and Salamon, in their monographs and papers, there are no effective method to establish transversality criteria in higher dimensional case. The deformation theory of pseudoholomorphic curves is governed by the so-called real Cauchy-Riemann operators(which are not the usual complex Cauchy-Riemann operators), we want to establish the Hormander－Kohn－Morrey theory for this kind of systems of PDEs, then we will consider applications to transversality criteria and super-rigidity of pseudoholomorphic curves, we also hope we will be able to find interesting applications to the study of entire holomorphic curves in complex manifolds(geometric hyperbolicity).