2004/10/11 by Yat-Ming Chan, Chan, Yat-Ming
Mathematics · #32Q25 #53C29 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:32Q25 #msc:53C29
paper · pdf · doi:10.48550/arxiv.math/0410260
29 pages
arxiv created 2004/10/11 · arxiv updated 2009/12/01
We study Calabi-Yau 3-folds M0 with a conical singularity x modelled on a Calabi-Yau cone V. We construct desingularizations of M0, obtaining a 1-parameter family of compact, nonsingular Calabi-Yau 3-folds which has M0 as the limit. The way we do is to choose an Asymptotically Conical Calabi-Yau 3-fold Y modelled on the same cone V, and then glue into M0 at x after applying a homothety to Y. We then get a 1-parameter family of nearly Calabi-Yau 3-folds Mt depending on a small real variable t. For sufficiently small t, we show that the nearly Calabi-Yau structures on Mt can be deformed to genuine Calabi-Yau structures, and therefore obtaining the desingularizations of M0. Our result can be applied to resolving orbifold singularities and hence provides a quantitative description of the Calabi-Yau metrics on the crepant resolutions.