Geometry of twisted Kähler-Einstein metrics and collapsing

Abstract

We prove that the twisted Kähler-Einstein metrics that arise on the base of certain holomorphic fiber space with Calabi-Yau fibers have conical-type singularities along the discriminant locus. These fiber spaces arise naturally when studying the collapsing of Ricci-flat Kähler metrics on Calabi-Yau manifolds, and of the Kähler-Ricci flow on compact Kähler manifolds with semiample canonical bundle and intermediate Kodaira dimension. Our results allow us to understand their collapsed Gromov-Hausdorff limits when the base is smooth and the discriminant has simple normal crossings.

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

“…The partial second-order estimate is motivated by the study of collapsing problems in Kähler geometry (see, e.g., [1,6,14,22,23,28,29,32,33,34,35,41,42,47,59,53,54,55,56,58,63,64,66,67,68,69,71,74,80]), as well as the study of canonical metrics in Kähler geometry and the behavior of the Kähler-Ricci flow. More specifically, the estimate in Theorem A plays a crucial role in establishing the following conjectural picture for collapsed Gromov-Hausdorff limits of Ricci-flat Kähler metrics which were originally proposed in [66,67], inspired by [24,38,39] and has been intensively studied since:…”

“…In [58], the Gromov-Hausdorff convergence was proved in general, and the homeomorphism type of the Gromov-Hausdorff limit was identified when the base has at worst orbifold singularities. In [23], following the program laid out in [71], the conjecture was proved when the base is log smooth.…”

“…The proof of Theorem B relies on the following 'novel' technique: In contrast with previous attempts to obtain the second-order estimate, we use the Chern-Lu inequality in place of the Aubin-Yau inequality. Hence, in place of using a metric with a lower bound on its bisectional curvature (c.f., [23,25]) we use a metric with a negative upper bound on its holomorphic sectional curvature (this was initially inspired by its use in [31], albeit Royden's version [49], which is what we use, is not realized in [31]). The motivation to forsaken the Aubin-Yau inequality in place of the Chern-Lu inequality is concentrated in the multiplicity of the divisorial pole |s E | −2ε .…”

“…The motivation to forsaken the Aubin-Yau inequality in place of the Chern-Lu inequality is concentrated in the multiplicity of the divisorial pole |s E | −2ε . Using the conical Aubin-Yau inequality established in [25], Gross-Tosatti-Zhang [23] were able to show that…”

“…In section 2, we give the proof of Theorem A. The proof of Theorem A being sufficient to establish Conjecture has been given in [1], and in [71,23] for the ε = 0 case. More precisely, [71,23] show that parts (i) and (ii) of the Conjecture can be established from the second-order estimate…”