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K-STABILITY AND PARABOLIC STABILITY
"... Abstract. Parabolic structures with rational weights encode certain iterated blowups of geometrically ruled surfaces. In this paper, we show that the three notions of parabolic polystability, K-polystability and existence of constant sca-lar curvature Kähler metrics on the iterated blowup are equiv ..."
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Abstract. Parabolic structures with rational weights encode certain iterated blowups of geometrically ruled surfaces. In this paper, we show that the three notions of parabolic polystability, K-polystability and existence of constant sca-lar curvature Kähler metrics on the iterated blowup are equivalent, for certain polarizations close to the boundary of the Kähler cone. 1.
Mass in Kähler Geometry
"... We prove a simple, explicit formula for the mass of any asymp-totically locally Euclidean (ALE) Kähler manifold, assuming only the sort of weak fall-off conditions required for the mass to actually be well-defined. For ALE scalar-flat Kähler manifolds, the mass turns out to be a topological invari ..."
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We prove a simple, explicit formula for the mass of any asymp-totically locally Euclidean (ALE) Kähler manifold, assuming only the sort of weak fall-off conditions required for the mass to actually be well-defined. For ALE scalar-flat Kähler manifolds, the mass turns out to be a topological invariant, depending only on the underly-ing smooth manifold, the first Chern class of the complex structure, and the Kähler class of the metric. When the metric is actually AE (asymptotically Euclidean), our formula not only implies a positive mass theorem for Kähler metrics, but also yields a Penrose-type in-equality for the mass. A complete connected non-compact Riemannian n-manifold (M, g) is said to be asymptotically Euclidean (or AE) if there is a compact subset K ⊂M such that M −K consists of finitely many components, each of which is dif-feomorphic to the complement of a closed ball Dn ⊂ Rn, in a manner such that g becomes the standard Euclidean metric plus terms that fall off (suffi-ciently rapidly) at infinity. More generally, a Riemannian n-manifold (M, g) is said to be asymptotically locally Euclidean (or ALE) if the complement of a compact set K consists of finitely many components, each of which is diffeomorphic to a quotient (Rn−Dn)/Γ, for some finite subgroup Γ ⊂ O(n) which acts freely on the unit sphere, in such a way that g again becomes the Euclidean metric plus error terms that fall off at infinity. The components of M −K are called the ends of M; their fundamental groups are the afore-mentioned groups Γ, which may in principle be different for different ends of the manifold.
