Symplectic Birational Geometry and Almost Complex Algebraic Geometry

辛双有理几何和近复代数几何

• 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

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.

几何学科始于希腊数学家欧几里得，他首先研究平面中的距离和角度之间的关系，然后研究空间中的距离和角度之间的关系。大约200年前，高斯和黎曼打开了现代几何学的大门。他们根据“流形”这一更一般的概念来研究几何。这是一个不一定是平坦的空间，尽管局部它类似于欧几里得空间，例如球体。他们研究的几何称为黎曼几何，它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，我们需要对黎曼几何进行修改，其中的局部模型是一个复平面而不是一个真实的几何。另一个概括是辛几何，其中我们将度量的概念（即距离和角度）更改为 2 形式。在平面上，它只是面积形式。辛几何的思想已经隐含在平面中。拉格朗日在分析力学方面的工作，以及后来在雅可比和汉密尔顿的经典力学公式中，赫尔曼·韦尔在他的《经典群》一书中首次使用了“辛”这个词。它源自一个希腊词。意思是“复杂”，这个词已经在数学中使用，具有不同的含义。在弦理论（一种为我们的宇宙提供可能模型的理论）的研究中，这两种几何学结合在一起提供了数学基础。该研究研究了辛流形的全局性质以及与复流形的相互作用。Enriques 和 Kodaira 描述了复曲面的双有理分类，即复 2-流形。表面根据其小平维数分为四类，其值为负无穷大、0、1 和 2。最小模型程序（Mori 程序）旨在将这些结果推广到更高维的复杂射影簇。该程序于 20 世纪 80 年代在 3 维中完成，并且最近已知可用于一般类型的复杂射影簇。辛拓扑是一门涉及辛流形的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 4 维中，也更加狂野。例如，任何有限呈现群都可以实现为辛 4-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需要研究。有两种自然的方法可以将双有理分类和双有理几何的其他方面扩展到辛流形。首先是修复辛结构。我们研究几何和拓扑在简单的双有理运算（例如辛放大/放大和辛变形）下如何变化。这称为辛双有理几何。该主题的技术和风格或多或少是拓扑性的，这提供了很大的灵活性。另一种方法是修复一个被辛形式驯服的近乎复杂的结构。这称为近复代数几何，比较严格。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 对偶性、圆锥定理和线性系统）推广到 4 维辛流形。来自不同学科的技术和相互作用，例如低维拓扑、代数几何、微分几何、复几何和辛拓扑对于这个项目非常重要。

项目成果

Moduli space of J-holomorphic subvarieties

J-全纯子族的模空间

• DOI: 10.1007/s00029-021-00648-z
• 发表时间: 2016-01-28
• 期刊: Selecta Mathematica
• 作者: Weiyi Zhang

Moduli space of $J$-holomorphic subvarieties

$J$-全纯子族的模空间

• DOI: http://dx.10.48550/arxiv.1601.07855
• 发表时间: 2016
• 期刊: arXiv e-prints
• 作者: Zhang Weiyi

Intersection of almost complex submanifolds

几乎复杂的子流形的交集

• DOI: http://dx.10.48550/arxiv.1707.08253
• 发表时间: 2017
• 作者: Zhang W

Intersection of almost complex submanifolds

几乎复杂的子流形的交集

• DOI: http://dx.10.4310/cjm.2018.v6.n4.a2
• 发表时间: 2018
• 期刊: Cambridge Journal of Mathematics
• 影响因子: 1.6
• 作者: Zhang W

$J$-holomorphic curves from closed $J$-anti-invariant forms

$J$-封闭 $J$-反不变形式的全纯曲线

• DOI: 10.4310/cag.2022.v30.n6.a1
• 发表时间: 2018-08-28
• 期刊: Communications in Analysis and Geometry
• 影响因子: 0.7
• 作者: Louis Bonthrone;Weiyi Zhang
• 通讯作者: Weiyi Zhang