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Symplectic Birational Geometry and Almost Complex Algebraic Geometry

辛双有理几何和近复代数几何

基本信息

批准号：
EP/N002601/1
负责人：
Weiyi Zhang
金额：
$ 12.68万
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2015
资助国家：
英国
起止时间：
2015 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FN002601%2F1
关键词：
Symplectic Birational Geometry Almost Complex

项目摘要

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.

几何学的主题始于希腊数学家欧几里德，他研究了距离和角度之间的关系，首先是在平面上，然后是在空间中。大约200年前，高斯和黎曼开启了现代几何学的大门。他们在更一般的“流形”概念上研究几何。这是一个不一定是平坦的空间，尽管在局部上它类似于欧几里得空间，例如球面。他们研究的几何称为黎曼几何，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体，其中局部模型是复杂平面而不是真正的平面。另一种推广是辛几何，我们将度量的概念，即距离和角度，改为2-形式。在飞机上，这只是面积形式。辛几何的思想已经隐含地出现在拉格朗日的分析力学著作中，后来又出现在雅各比和汉密尔顿的经典力学公式中。赫尔曼·韦尔在他的《古典群》一书中首次使用了辛词。它源于希腊语中的一个词，意思是复杂，这个词在数学中已经使用了不同的含义。在弦理论的研究中，弦理论为我们的宇宙提供了一个可能的模型，这两个几何结合在一起提供了数学基础。研究辛流形的整体性质及其与复流形的相互作用。Enriques和Kodaira描述了复杂曲面的双分类，即复2-流形。根据它们的Kodaira维度，曲面被分为四类，取值为负无穷大，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射影变体。该程序是在20世纪80年代在3维空间上完成的，最近已被用于一般类型的复射影变种。辛拓扑是关于辛流形的一个重要的整体问题。与复流形相比，辛流形的拓扑，即使是在4维的情况下，也要狂野得多。例如，任何有限表示的群都可以实现为辛四维流形的基本群。因此，在辛拓扑中，我们有比复流形更多的目标要研究。有两种自然的方法将双态分类和双态几何的其他方面推广到辛流形。第一个是确定辛结构。我们研究了几何和拓扑在简单的双子运算下是如何变化的，如辛的爆破/爆破和辛形变。这就是辛二次几何。这个主题的技巧和风格或多或少都是拓扑性的，这给了我们很大的灵活性。另一种方法是修正被辛形式驯服的几乎复杂的结构。这被称为几乎复代数几何，它更严格。我们计划利用J-全纯曲线的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。低维拓扑、代数几何、微分几何、复几何和辛拓扑等不同学科的技术和相互作用对这个项目非常关键。