Applications of Polyhedral Kahler Manifolds.

得到具有全纯结构的流形有许多不同的方法。在实维2的情况下，有向表面上的任何黎曼度规都在其上定义了一个复杂结构。在高维空间中，我们可以利用代数几何，在CP^n中取由几个代数超曲面的交给出的子流形，并在其上推导出由CP^n导出的全纯结构。多面体卡勒流形是由一种不同的，在某种意义上更组合的构造得到的复杂流形。这些流形是在我的博士论文中介绍的，我记得它们的定义。考虑一个维数为2n的流形，并对其进行简单分解，在每个简单形上选择一个欧氏度规。这定义了一个具有余维为2的圆锥奇点的平坦度规。考虑流形的非奇异部分上度规的完整性。如果度量的完整性包含在SO（2n）的子群U(n)中，则称其为多面体Kahler （PK）。结果表明，每一个PK流形都是一个复流形，度规的奇异性在其上形成一个（通常是约简的）全纯因子。关于PK参数的主要问题如下。给定一个复杂流形M^n，是否有可能找到除数D1，…， Dk使得M^n上存在一个PK度规，它恰好沿着除数D1，…, Dk ?是否存在不允许有PK度规的代数复曲面是未知的。但是对于大多数在>1维流形上构造的PK度量，度量是刚性的，没有模。度规的存在导致在除数上的上同调方程的系统，它可以采取以下形式：在CP^2上对3n条线的排列进行分类，使得每条线与其他线正好相交于n+1个点。对于所有这样的排列（n>1），在CP^2上存在一个沿其有奇点的PK度规。构造具有大基群的紧致复流形是困难的。具有可收缩泛覆盖的紧复流形的例子很少。多面体Kahler度量可以用来构造这样的例子。构造非正曲率度规就足够了。这些关于负曲率的想法使我得到了以下关于CP^2猜想的主要问题的推测性部分答案。考虑CP^2中的一种线排列，它是一个PK度规的奇异轨迹。那么它的补就是K（,1）型。我想用最小曲面来证明这个猜想。具有完全非正曲率度规的空间中的极小曲面具有非正曲率，因此不可能是2球。补线排列上的PK度量是平的，但不是完整的。因此，一般地，使所述排列补片中的面积最小的一系列表面收敛于沿曲线接触所述排列的分段光滑表面。据推测，极限曲面上的度规是C^1光滑的，并且在排列的多个点上具有圆锥奇点，否则它具有非正曲率。而且所有圆锥形点的角都大于2。因此，曲面不可能是球面，并且排列补的pi(2)必须为零。由于补体可收缩为二维细胞复合体，因此它必须是K（pi,1）型。这个项目的一个更远大的目标是证明以下关于复杂反射排列的经典猜想。设V为有限维复向量空间，GL(V)中的W为有限维复反射群。反射超平面在V中的补是一个K（,1）空间。根据Couwenberg、Heckman和Looijenga的研究，复杂反射排列的投影通常是CP^n上多面体Kahlermetric的奇异集。因此，上述观点可以进一步发展，以证明这一猜想。

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.