“…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:…”