Applications of Polyhedral Kahler Manifolds.

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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.