Applications of Polyhedral Kahler Manifolds.

有许多不同的方法可以得到具有全纯结构的流形。在实数维为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数结构。在高维空间中，人们可以利用代数几何，取Cp^n中由几个代数超曲面的交集给出的子流形，并在其上诱导出从Cp^n到Cp^n的全纯结构。多面体Kahler流形是通过不同的，在某种意义上更组合的构造得到的复流形。这些流形是在我的博士论文中引入的，我还记得那个定义。考虑一个具有单纯分解的2n维流形，并在每个单纯形上选择一个欧几里得度量。这定义了一个具有余维2的锥形奇点的平坦度量。考虑流形的非奇异部分上的度量的完整。如果度量包含在SO(2n)的子群U(N)中，则称该度量为多面体Kahler(PK)。证明了每个PK流形都是复流形，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，是否有可能在其上找到因子d1，…，dk，使得在M^n上存在一个pk度量，它精确地沿着除数d1，…，dk有奇点？是否存在一个不允许pk度量的代数复曲面是未知的。但是对于&gt；1维流形上的大多数构造的PK度量，度量是刚性的，并且没有模。度量的存在导致了一个关于因子的上同调方程组，它可以采取以下形式：问题。对Cp^2上3n条直线的排列进行了分类，使得每条直线在n+1点处精确地与其他直线相交.对于所有这样的排列(n&gt；1)，在Cp^2上存在沿它们的奇点的PK度量，很难构造具有大基本群的紧致复流形.已知的具有可压缩泛覆盖的紧致复流形的例子很少。多面体Kahler度量可以用来构造这样的例子。构造非正曲率的度量就足够了。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以下猜想的部分答案。考虑Cp^2中的一条线排列，它是PK度规的奇异轨迹。那么它的补集是K(pi，1)型的。我想用极小曲面证明这个猜想。具有非正曲率完备度量的空间中的极小曲面具有非正曲率，因此不可能是2-球面。行排列的补码上的PK度量是平坦的，但不是完全的。因此，一般而言，使该布置的补集中的面积最小的一系列曲面收敛到沿曲线接触该布置的分段光滑表面。猜想极限曲面上的度量是C^1光滑的，并且在排列的多个点处具有圆锥奇点，否则它具有非正曲率。此外，所有圆锥点的夹角都大于2pi。因此，该曲面不能是球面，并且该排列的补集的pi(2)必须为零。由于补码是可收缩到2维单元复形的，所以它一定是K(pi，1)型的。这个项目更雄心勃勃的目标是证明下面关于复反射排列的经典猜想。设V是有限维复向量空间，GL(V)中的W是有限复反射群。由Couwenberg，Heckman和Looijenga的工作得出，复杂反射排列的射影通常是Cp^n上多面体Kahler距离的奇集，从而可以进一步发展上述思想来证明这一猜想。

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.