Symplectic Birational Geometry and Almost Complex Algebraic Geometry

几何学始于希腊数学家欧几里得，他研究距离和角度之间的关系，首先是在平面上，然后是在空间上。大约200年前，高斯和黎曼打开了现代几何的大门。他们在更普遍的“流形”概念上学习几何。这是一个不一定是平的空间，尽管局部它像欧几里得空间，例如球体。他们研究的几何被称为黎曼几何，这是爱因斯坦广义相对论的数学基础。在物理学的研究中，人们发现，在某些情况下，我们需要修改黎曼几何。一个方向是复杂几何，其中局部模型是一个复杂平面而不是一个实平面。另一个推广是辛几何，我们将度量的概念，即距离和角度，改为2形式。在平面上，它只是面积形式。辛几何的概念在拉格朗日关于分析力学的著作中，以及后来在雅可比和汉密尔顿关于经典力学的表述中，已经隐式地出现了。赫尔曼·魏尔在他的著作《古典群》中首次使用了“辛”这个词。它来源于一个希腊词，意思是复杂的，这个词在数学中已经有了不同的含义。弦理论是一种为我们的宇宙提供可能模型的理论，在弦理论的研究中，这两种几何结合在一起提供了数学基础。本文主要研究辛流形的整体性质及其与复流形的相互作用。Enriques和Kodaira描述了复曲面的两族分类，即复2流形。这些表面根据它们的Kodaira尺寸分为四类，分别取负无穷、0、1和2。最小模型程序（Mori程序）旨在将这些结果推广到高维复杂投影品种。该程序于20世纪80年代在三维空间完成，最近已知可用于一般类型的复杂投影品种。辛拓扑是一门涉及辛流形重要全局问题的学科。与复流形相比，辛流形的拓扑结构，即使在第4维，也要复杂得多。例如，任何有限表示的群都可以被实现为辛4流形的基本群。因此，在辛拓扑中，我们有比复杂流形更多的研究目标。将二元分类和二元几何的其他方面扩展到辛流形有两种自然的方法。第一种是固定一个辛结构。我们研究了几何结构和拓扑结构在简单的二元操作下的变化，如辛膨胀/膨胀和辛变形。这被称为辛二分几何。这个主题的技术和风味或多或少是拓扑的，这给了很大的灵活性。另一种方法是固定一个由辛形式驯服的几乎复杂的结构。这被称为几乎复杂的代数几何，刚性更强。我们计划利用j -全纯曲线理论将代数几何的相关部分（特别是Nakai-Moishezon和Kleiman对偶、锥定理和线性系统）推广到4维辛流形。来自不同学科的技术和相互作用，如低维拓扑、代数几何、微分几何、复杂几何和辛拓扑，对这个项目非常重要。

The subject of geometry begins with the Greek mathematician Euclid who studied relationships among distances and angles, first in a plane and then in a space. About 200 years ago, Gauss and Riemann opened the door of modern geometry. They studied geometry on the more general notion of "manifold''. This is a space which is not necessarily flat, although locally it is like an Euclidean space, e.g. a sphere. The geometry studied by them is called Riemannian geometry, which is the mathematical foundation of Einstein's general relativity. In the study of Physics, people find that, in some situations, we need modifications of Riemannian geometry. One direction is complex geometry, where the the local model is a complex plane instead of a real plane. Another generalization is symplectic geometry, where we change the notion of metrics, i.e. distances and angles, to a 2-form. On a plane, it is just the area form. The idea of symplectic geometry made an implicit appearance already in the work of Lagrange on analytical mechanics and later in Jacobi's and Hamilton's formulation of classical mechanics. It is Herman Weyl who first uses the word it symplectic in his book Classical Groups. It is derived from a Greek word meaning complex, a word already used in mathematics with a different meaning. In the study of String Theory, a theory providing a possible model for our universe, these two geometries come together to provide mathematical foundations. The proposed research studies the global property of symplectic manifolds and the interactions with complex manifolds.Enriques and Kodaira described the birational classification of complex surfaces, i.e. complex 2-manifolds. The surfaces are divided into four categories according to their Kodaira dimensions, which take values negative infinity, 0, 1, and 2. The Minimal Model Program (Mori program) aims to generalize these results to higher dimensional complex projective varieties. This program is complete in dimension 3 in 1980s and is known to work for complex projective varieties of general type recently. Symplectic topology is a subject concerning important global questions of symplectic manifolds. Comparing to complex manifolds, the topology of symplectic manifolds, even in dimension 4, is far more wild. For example, any finitely presented group can be realized as the fundamental group of a symplectic 4-manifold. Hence in symplectic topology, we have many more objectives to study than complex manifolds.There are two natural ways to extend the birational classification and other aspects of birational geometry to symplectic manifolds. The first is to fix a symplectic structure. We study how the geometry and topology are changing under simple birational operations like the symplectic blow-up/blow-down and symplectic deformations. This is called the symplectic birational geometry. The techniques and flavours of this subject are more or less topological which gives a lot of flexibility. The other way is to fix an almost complex structure tamed by a symplectic form. This is called the almost complex algebraic geometry, which is more rigid. We plan to use the theory of J-holomorphic curves to generalize the relevant part of algebraic geometry (in particular the Nakai-Moishezon and Kleiman dualities, the cone theorem and linear systems) to symplectic manifolds of dimension 4.Techniques and interactions from different disciplines, e.g. low dimensional topology, algebraic geometry, differential geometry, complex geometry and symplectic topology, are very crucial for this project.