Constant $$\mu $$-Scalar Curvature Kähler Metric—Formulation and Foundational Results

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 ...

Hypersurfaces with Constant Scalar Curvature

Let M be a complete two-dimensional surface immersed into the three-dimensional Euclidean space. Then a classical theorem of Hilbert says that when the curvature of M is a non-zero constant, M must be the sphere. On the other hand, when the curvature of M is zero, a theorem of Har tman-Nirenberg [4] says that M must be a plane or a cylinder. These two theorems complete the classification of com...

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

Blowing up and Desingularizing Constant Scalar Curvature Kähler Manifolds

This paper is concerned with the existence of constant scalar curvature Kähler metrics on blow ups at finitely many points of compact manifolds which already carry constant scalar curvature Kähler metrics. We also consider the desingularization of isolated quotient singularities of compact orbifolds which already carry constant scalar curvature Kähler metrics. Let (M,ω) be either a m-dimensiona...