On the degeneration of asymptotically conical Calabi-Yau metrics

We study the degenerations of asymptotically conical Ricci-flat Kahler metrics as the Kahler class degenerates to a semi-positive class. We show that under appropriate assumptions, the Ricci-flat Kahler metrics converge to a incomplete smooth Ricci-flat Kahler metric away from a compact subvariety. As a consequence, we construct singular Calabi-Yau metrics with asymptotically conical behaviour at infinity on certain quasi-projective varieties and we show that the metric geometry of these singular metrics are homeomorphic to the topology of the singular variety. Finally, we will apply our results to study several classes of examples of geometric transitions between Calabi-Yau manifolds.