论文标题
从光滑到几乎复杂
From Smooth to Almost Complex
论文作者
论文摘要
本文主要旨在概述为开发代数几何形状以任意紧凑的几乎复杂流形的最新努力。 我们回顾了指导哲学获得的结果，即从雷内·托姆（RenéThom）的横向性方面，平滑歧管之间平滑地图的陈述也应在伪酚晶体环境中具有对应物，而不需要几乎复杂的结构的一般性。这些包括紧凑的几乎复杂的亚曼福尔德的相交，伪晶图的结构，某些谐波形式的零位点以及Laplacian的特征值。除了审查紧凑型歧管情况外，我们还将这些结果扩展到了Orbifolds和非压缩歧管。讨论了动机，方法论，应用和进一步的方向。 假酚形态图上的结构结果导致了几乎复杂的歧管之间的生育形态的概念。在这种情况下，这激发了对包括Kodaira尺寸和Plurigenera在内的各种男性不变的研究。还审查了几乎复杂的代数几何形状的其他一些方面$ 4 $。这些包括球形类别中的（CO）同源类别和亚Varieties的锥体。
This article mainly aims to overview the recent efforts on developing algebraic geometry for an arbitrary compact almost complex manifold. We review the results obtained by the guiding philosophy that a statement for smooth maps between smooth manifolds in terms of René Thom's transversality should also have its counterpart in pseudoholomorphic setting without requiring genericity of the almost complex structures. These include intersection of compact almost complex submanifolds, structure of pseudoholomorphic maps, zero locus of certain harmonic forms, and eigenvalues of Laplacian. In addition to reviewing the compact manifolds situation, we also extend these results to orbifolds and non-compact manifolds. Motivations, methodologies, applications, and further directions are discussed. The structural results on the pseudoholomorphic maps lead to a notion of birational morphism between almost complex manifolds. This motivates the study of various birational invariants, including Kodaira dimensions and plurigenera, in this setting. Some other aspects of almost complex algebraic geometry in dimension $4$ are also reviewed. These include cones of (co)homology classes and subvarieties in spherical classes.
