Uniform Stability of Twisted Constant Scalar Curvature Kähler Metrics

F eb 2 00 5 AN OBSTRUCTION TO THE EXISTENCE OF CONSTANT SCALAR CURVATURE KÄHLER METRICS

We prove that polarised manifolds that admit a constant scalar curvature Kähler (cscK) metric satisfy a condition we call slope semistability. That is, we define the slope µ for a projective manifold and for each of its subschemes, and show that if X is cscK then µ(Z) ≤ µ(X) for all subschemes Z. This gives many examples of manifolds with Kähler classes which do not admit cscK metrics, such as ...

Constant scalar curvature metrics with isolated singularities

We extend the results and methods of [6] to prove the existence of constant positive scalar curvature metrics g which are complete and conformal to the standard metric on S \ Λ, where Λ is a disjoint union of submanifolds of dimensions between 0 and (N − 2)/2. The existence of solutions with isolated singularities occupies the majority of the paper; their existence was previously established by...

Constant scalar curvature metrics on toric surfaces

On the existence of Kähler metrics of constant scalar curvature

For certain compact complex Fano manifolds M with reductive Lie algebras of holomorphic vector fields, we determine the analytic subvariety of the second cohomology group of M consisting of Kähler classes whose Bando-Calabi-Futaki character vanishes. Then a Kähler class contains a Kähler metric of constant scalar curvature if and only if the Kähler class is contained in the analytic subvariety....

Constant Scalar Curvature Metrics on Boundary Complexes of Cyclic Polytopes

In this paper we give examples of constant scalar curvature metrics on piecewise-flat triangulated 3-manifolds. These types of metrics are possible candidates for “best” metrics on triangulated 3-manifolds. In the pentachoron, the triangulation formed by the simplicial boundary of the 4-simplex, we find that its stucture is completely deterimed with a vertex transitive metric. Further this metr...