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Applications of Polyhedral Kahler Manifolds.

多面体卡勒流形的应用。

基本信息

批准号：
EP/E044859/1
负责人：
Dmitri Panov
金额：
$ 31.1万
依托单位：
Imperial College London
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2007
资助国家：
英国
起止时间：
2007 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FE044859%2F1
关键词：
Applications Polyhedral Kahler Manifolds.

项目摘要

There exist many different ways to obtain manifolds with holomorphic structure. In the case of real dimension 2 any Riemannian metric on an oriented surface defines a complex structure on it. In higher dimensions one can use algebraicgeometry, take a submanifold in CP^n given by the intersection of several algebraic hypersurfaces and induce on it the holomorphic structure from CP^n. Polyhedral Kahlermanifolds are complex manifolds that are obtained by a different,and in some sense more combinatorial, construction. These manifolds were introduced in my Phd and I recall the definition.Consider a manifold of dimension 2n with a simplicial decomposition and choose a Euclidean metric on every simplex. This defines a flat metric with conical singularities of codimension 2. Consider the holonomy of the metric on the nonsingular part of themanifold. The metric is called polyhedral Kahler (PK) if its holonomy is contained in the subgroup U(n) of SO(2n). It turns out that every PK manifold is a complex manifold and the singularities of the metric form a (usually reduced) holomorphic divisor on it.The MAIN QUESTION about PK metrics is the following. Given a complex manifold M^n, is it possible to find divisors D1,..., Dk on it such that there exists a PK metric on M^n that has singularities precisely along the divisor D1,..., Dk?It is unknown if there exits an algebraic complex surface that doesn't admit a PK metric. But for the majority of constructed PK metrics on manifolds of dimension >1 the metric is rigid and has no moduli. The existence of the metric leads to a system of cohomological equations on divisors that can take the following form:Problem. Classify arrangements of 3n lines on CP^2 such that every line intersect other lines exactly at n+1 points.For all such arrangements (n>1) there exists a PK metric on CP^2 with singularities along them.It is hard to construct compact complex manifolds with large fundamental group. Very few examples of compact complex manifold with contractible universal covering are known. Polyhedral Kahler metrics can be used to construct such examples. It is sufficient to construct metric of non-positive curvature. These ideas about negaive curvature lead me to the following conjectural partial answer to the MAIN QESTION in case of CP^2.Conjecture. Consider a line arrangement in CP^2 that is a singular locus of a PK metric. Then its complement is of type K(pi,1).I want to prove this conjecture using minimal surfaces. A minimal surface in a space with a complete metric of non-positive curvature has non positive curvature and socan not be a 2-sphere. The PK metric on the complement to the line arrangement is flat but is not complete. Thus generically a sequences of surfaces minimizing the area in the complement to the arrangement converges to a piecewise smooth surface that touches the arrangement along a curve. Conjecturally the metric on the limiting surface is C^1 smooth and have conical singularities at multiple points of the arrangement and otherwise it has non-positive curvature. Moreover angles of all conical points aregreater than 2pi. Thus the surface can not be a sphere and pi(2) of the complement to the arrangement must be zero. Since the complement is contractible to a 2-dimensional cell complex it must be of type K(pi,1).A more ambitious goal of the project is to prove the following classical conjecture about complex reflection arrangements.Conjecture. Let V be a finite dimensional complex vector space and W in GL(V) be a finite complex reflection group. The complement in V of the reflecting hyperplanes is a K(pi,1) space.It follows from the work of Couwenberg, Heckman, and Looijenga that the projectivization of a complex reflection arrangement is often a singular set of a polyhedral Kahlermetric on CP^n. Thus the ideas above could be developed further to prove this conjecture.

有许多不同的方法来获得具有全纯结构的流形。在真实的维数为2的情况下，定向曲面上的任何黎曼度量都定义了其上的复结构，在高维情况下，人们可以利用代数几何，取CP^n中由几个代数超曲面的交给出的子流形，并在其上导出CP^n的全纯结构。多面体Kahlermanifold是复杂的流形，通过不同的，在某种意义上更组合，建设。这些流形是在我的博士论文中介绍的，我记得它们的定义。考虑一个具有单纯分解的2n维流形，在每个单纯形上选择一个欧几里得度量。这定义了一个平坦的度量与圆锥奇点的余维2。考虑流形非奇异部分上度量的完整性。如果度量的完整性包含在SO（2n）的子群U（n）中，则称度量为多面体Kahler（PK）.证明了每个PK流形是一个复流形，度量的奇点在其上形成一个（通常是约化的）全纯因子。关于PK度量的主要问题如下。给定一个复流形M^n，是否有可能找到因子D1，...，Dk，使得在M^n上存在PK度量，其具有精确地沿着除数D1，.，沿着的奇点，Dk？不知道是否存在不允许PK度量的代数复曲面。但对于维数>1的流形上的大多数构造的PK度量，度量是刚性的，没有模。度量的存在导致了一个关于因子的上同调方程系统，它可以采取以下形式：问题。对CP^2上的3 n条直线的排列进行分类，使得每条直线与其它直线相交于n+1点，对于所有的排列（n>1），CP^2上都存在一个PK度量，且沿着它们有沿着奇点.构造具有大基本群的紧致复流形是困难的.已知的紧致复流形具有可收缩泛覆盖的例子很少。多面体Kahler度量可以用来构造这样的例子。构造非正曲率的度量是足够的。这些关于负曲率的想法使我对CP ^[2]猜想中的主要问题给出了下面的几何上的部分回答。考虑CP ^2中的一个线排列，它是PK度量的一个奇异轨迹。那么它的补是K（pi，1）型的。我想用极小曲面证明这个猜想。具有非正曲率的完备度量的空间中的极小曲面具有非正曲率，因而不能是2-球面。关于线路布置的补充的PK度量是平坦的，但不完整。因此，一般地，最小化该布置的互补中的面积的表面序列收敛于沿沿着曲线接触该布置的分段平滑表面。假设极限曲面上的度规是C^1光滑的，并且在多个点上有圆锥奇点，否则它有非正曲率。而且所有锥点的夹角都大于2 π。因此曲面不可能是球面，并且排列的补曲面的pi（2）必须为零。由于补是可收缩的二维细胞复合体，它必须是类型K（pi，1）。该项目的一个更雄心勃勃的目标是证明以下关于复反射的经典猜想。设V是有限维复向量空间，GL（V）中的W是有限复反射群. V中反射超平面的补空间是K（pi，1）空间，从Couwenberg、Heckman和Looijenga的工作可以得出，复反射排列的射影化通常是CP^n上多面体Kahlermetric的奇异集。因此，上述思想可以进一步发展，以证明这一猜想。