Since Calabi's original paper, the Calabi Ansatz has been central for constructions in Kähler geometry. Calabi himself used it to construct complete Ricci-flat metrics on the total space of the canonical bundle of a Kähler-Einstein Fano manifold (B, \omega_B), generalizing some well-known examples coming from physics. Over the years, work of Koiso, Cao, Feldman-Ilmanen-Knopf, Futaki-Wang, Chi Li, and others have shown that the Calabi Ansatz can be used to produce complete Kähler-Ricci solitons, important singularity models for the Kähler-Ricci flow, on certain line bundles over (B, \omega_B). In this talk, I'll explain a generalization of these results to the total space of some higher-rank direct-sum vector bundles over (B, \omega_B). In our case the Calabi Ansatz is not suitable, and we instead use the theory of hamiltonian 2-forms, introduced by Apostolov-Calderbank-Gauduchon-Tønneson-Friedman. The construction produces new examples of complete shrinkers, steadies, and Calabi-Yau metrics, as well as recovering some known ones.

Ricci-flat metrics are fundamental in differential geometry, and they are easier to study when they have additional structures. I will describe my works on 4d non-trivially conformally Kähler Ricci-flat metrics, which actually is a very natural class of 4d Ricci-flat metrics. This leads to a classification of asymptotic geometries of such metrics at infinity and a classification of such gravitational instantons

In this joint work with Mihai Paun, we show that a so-called Q-sheaf on a log terminal Kahler threefold satisfies a suitable Bogomolov-Gieseker inequality as soon as it is stable with respect to a Kahler class. I will discuss the strategy of the proof as well as some applications if time permits.

Steady Kahler-Ricci solitons are eternal solutions of the Kahler-Ricci flow. I will present new examples of such solitons with strictly positive sectional curvature that live on C^n and provide an answer to an open question of H.-D. Cao in complex dimension 2. This is joint work with Pak-Yeung Chan and Yi Lai.

Gravitational instantons are non-compact Calabi-Yau metrics with L^2 bounded curvature and are categorized into six types. We will focus on the ALH* type which has a non-compact end with inhomogeneous collapsing near infinity. I will talk about a joint project with Rafe Mazzeo on the Fredholm mapping property of the Laplacian and the Dirac operator, where the geometric microlocal analysis of fibered metrics plays a central role. Application of this Fredholm theory includes the L^2 Hodge theory, polyhomogeneous expansion and local perturbation theory. I will also discuss a joint project with Yu-Shen Lin and Sidharth Soundararajan on the degeneration of such metrics which gives a partial compactification of their moduli space.