Conic Kähler–Einstein metrics along simple normal crossing divisors on Fano manifolds

Abstract

We prove that on one Kähler-Einstein Fano manifold without holomorphic vector fields, there exists a unique conical Kähler-Einstein metric along a simple normal crossing divisor with admissible prescribed cone angles. We also establish a curvature estimate for conic metrics along a simple normal crossing divisor which generalizes Li-Rubinstein's curvature estimate for one divisor case.

Second-order estimates for collapsed limits of Ricci-flat Kähler metrics

“…From [19, 22], we know that there is a uniform constant such that . This, in particular, implies that the holomorphic sectional curvature of the cone metric affords the (positive) upper bound for .…”

“…Second-Order Estimates for Collapsed Limits of Ricci-flat Kähler Metrics 3 but is bounded above (see [19,22,26]). Even in the case of a single smooth divisor, the computation in [19] shows that the holomorphic sectional curvature of the reference conical metric decays to −∞ near the divisor.…”

“…There, the Aubin–Yau inequality is used, making use of the curvature lower bound on the reference conical metric obtained by Guenancia and Paun [16]. In general, the curvature of the reference conical metric is not bounded below, but is bounded above (see [19, 22, 26]). Even in the case of a single smooth divisor, the computation in [19] shows that the holomorphic sectional curvature of the reference conical metric decays to near the divisor.…”

The Partial $C^{0}$-estimate along a general continuity path and applications

“…Another important ingredient in the study of Kähler-Einstein problem is the conic Kähler metrics. As a natural generalization of Kähler-Einstein metrics, the conical Kähler-Einstein metrics were studied in [3,18,19,21,24,26,38,39] and played the important role in the solution to the Yau-Tian-Donaldson Conjecture. In fact the conical Kähler-Einstein metrics with deforming cone angles give rise to the continuity path which establishes the existence of the smooth Kähler-Einstein metric as soon as cone angle attains 2π.…”