Higher regularity for singular Kähler-Einstein metrics

Abstract

We study singular Kähler-Einstein metrics that are obtained as non-collapsed limits of polarized Kähler-Einstein manifolds. Our main result is that if the metric tangent cone at a point is locally isomorphic to the germ of the singularity, then the metric converges to the metric on its tangent cone at a polynomial rate on the level of Kähler potentials. When the tangent cone at the point has a smooth cross section, then the result implies polynomial convergence of the metric in the usual sense, generalizing a result due to Hein-Sun. We show that a similar result holds even in certain cases where the tangent cone is not locally isomorphic to the germ of the singularity. Finally we prove a rigidity result for complete ∂ ∂-exact Calabi-Yau metrics with maximal volume growth. This generalizes a result of Conlon-Hein, which applies to the case of asymptotically conical manifolds.Theorem 1.2. Suppose that, as above, (Z, p) is the pointed Gromov-Hausdorff limit of a non-collapsing sequence of polarized Kähler-Einstein manifolds, with singular Kähler-Einstein metric ω Z . Suppose that the germ (Z, p) is isomorphic to the germ (A p−1 , 0) at the origin. Then for some r 0 > 0 there is a biholomorphism φ : B(0, r 0 ) → U ⊂ Z, with φ(0) = p, and constants Λ, C, α > 0, such thatfor some u r defined on B(0, r), and sup B(0,r) |u r | ≤ Cr 2+α

Some models for bubbling of (log) Kähler–Einstein metrics

“…If β ∈ (5/6, 1) then [13] produce a Calabi-Yau metric g CY in a neighbourhood of 0 ∈ C 2 with cone angle 2πβ along C \ {0} whose tangent cone at the origin is equal to the product C × C γ where γ = 2β − 1. Following [9] we expect that g K E has polynomial convergence to (a multiple) of g CY at the level of potentials.…”