This talk focuses on the recent resolutions of several well-known conjectures in studying the Einstein 4-manifolds with special holonomy. The main results include the following.

(1) Any volume collapsed limit of unit-diameter Einstein metrics on the K3 manifold is isometric to one of the following: the quotient of a flat 3D torus by an involution, a singular special Kaehler metric on the topological 2-sphere, or the unit interval.

(2) Any complete non-compact hyperkaehler 4-manifold with quadratically integrable curvature, namely gravitational instanton, must have an ALX model geometry with optimal asymptotic rate.

(3) Any gravitational instanton is biholomorphic to a dense open subset of some compact algebraic surface.

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Last updated: 31 May 2023