Orbifold regularity of weak Kähler-Einstein metrics

Abstract

In the resolution of the YTD conjecture on the existence of Kähler-Einstein metrics on Fano manifolds (see [23] and also [5]), a crucial tool is a compactness result.In its simplest form, this result says that the Gromov-Hausdorff limit of a sequence of smooth Kähler-Einstein manifolds (X i , ω i,KE ) is a normal Fano variety X := X ∞ with klt singularities and that there is a weak Kähler-Einstein metric ω w ∞,KE on X ∞ .The existence of a Gromov-Hausdorff limit follows from Gromov's compactness theorem.So the important information in this statement is about the regularity of X ∞ .It was the second author ([20], [22], see also [15]) who first pointed out the route to prove that X ∞ is an algebraic variety is to establish a so-called partial C 0 -estimate.He demonstrated in [20] how to achieve this when the complex dimension n is equal to 2 by showing that a sequence of Kähler-Einstein surfaces converges to a Fano orbifold with a smooth orbifold Kähler-Einstein metric.Note that when n = 2, klt singularities are nothing but quotient singularities or orbifold singularities.Two key ingredients to prove the partial C 0 -estimate in dimension 2 are orbifold compactness result of Einstein 4-manifolds and Hörmander's L 2 -estimates.Recently, Donaldson-Sun [7] and the second author [21] generalized the partial C 0 -estimate to higher dimensional Kähler-Einstein manifolds.Here they need to rely on compactness results of higher dimensional Einstein manifolds developed by Cheeger-Colding and Cheeger-Colding-Tian (see [4] and the reference therein).Compared to the complex dimension 2 case, the second author also conjectured that ω ∞,KE is a smooth orbifold metric away from analytic subvarieties of complex codimension 3. Note that in [4], it was proved that the (metric) singular set of X ∞ has complex codimension at least 2.It can be shown that, by partial C 0 -estimate, there is a uniform C 2 -estimate of the potential of ω w ∞,KE on X reg ∞ .Then the Evans-Krylov theory or Calabi's 3rd derivative estimate allows one to show that ω w ∞,KE is smooth on]). Alternatively using Pǎun's Laplacian estimate in [16] and Evans-Krylov theory, Berman-Boucksom-Eyssidieux-Guedj-Zeriahi [1] showed directly that any weak Kähler-Einstein metric ω w KE on a klt Fano variety X ∞ is smooth on X reg ∞ .The purpose of this note is to answer the question by the second author about the regularity of ω w KE on the orbifold locus X orb ∞ of X ∞ .First, if (X, -K X ) is a klt Fano variety, then by [8, Proposition 9.3] there exists a closed subset Z ⊂ X with codim X Z ≥ 3 such that X\Z has quotient singularities.So we just need to show the following regularity result.For the definition of weak Kähler-Einstein metric, see Definition 1.

Locations

• Contemporary mathematics - American Mathematical Society - View
• arXiv (Cornell University) - View - PDF