Desingularizations of Calabi–Yau 3-folds with a conical singularity

This is the second of two papers studying Calabi–Yau 3-folds with conical singularities and their desingularizations. In our first paper [Y.-M. Chan, Quart. J. Math. 57 (2006), 151–181] we con-structed the desingularization of the conically singular manifold M0 by gluing an asymptotically conical (AC) Calabi–Yau 3-fold Y into M0 at the singular point, thus obtaining a 1-parameter family of compact, non-singular Calabi–Yau 3-folds Mt for small t> 0. During the gluing process one may encounter a kind of cohomological obstruction to defining a 3-form t on Mt which interpolates between the 3-form 0 on M0 and the scaled 3-form t3Y on Y if the rate λ at which the AC Calabi–Yau 3-fold Y converges to the Calabi–Yau cone is equal to −3. The first paper [3] studied the simpler case λ < −3 where there is no obstruction. This paper extends the result in the first one by considering a more complicated situtation when λ = −3. Assuming the existence of singular Calabi–Yau metrics on compact complex 3-folds with ordinary double points, our result in this paper can be applied to repairing such kinds of singularities, which is an analytic version of Friedman’s result giving necessary and sufficient conditions for smoothing ordinary double points. 1