Geometry on the Finite Time Collapsing for Continuity Method

This paper is concerned about a class of finite time collapsing along continuity method over a compact Kähler manifold. It is shown that along the continuity method the diameter is uniformly bounded and its Gromov–Hausdorff limit is homeomorphic and locally isometric to the regular part of base space. Furthermore, we also prove that the Gromov–Hausdorff limit is unique under the algebraic assumption of the Kähler class by applying the recent work of Song–Tian–Zhang [2019, arXiv:1904.08345].