Deforming Calabi-Yau Threefolds
Mark Gross
TL;DR
This work generalizes the Bogomolov–Tian–Todorov unobstructedness to Calabi–Yau threefolds with canonical singularities by showing that deformation obstructions are localized to singularities via a $T^1$–$T^2$ framework and RS-type lifting theory. It then derives smoothing results for singular CYs: isolated complete intersection singularities admit smoothings that remove all singularities except possible ordinary double points, and certain non-CI singularities are smoothable under a strong 'good' hypothesis; when crepant resolutions exist, the deformation spaces behave unobstructedly, and primitive contractions yield explicit smoothing criteria. The paper also provides a detailed treatment of complete intersection and non-complete intersection cases, as well as a classification of primitive contractions with precise smoothability conditions tied to the geometry of exceptional divisors. Together, these results illuminate the landscape of degenerations and smoothings of CY threefolds, enabling systematic construction of new CYs from singular models and informing their deformation theory. The methods synthesize local-to-global deformation theory, Ext-duality, and the geometry of resolutions and contractions to produce practical smoothing criteria and a richer structural understanding of CY moduli.
Abstract
This paper first generalises the Bogomolov-Tian-Todorov unobstructedness theorem to the case of Calabi-Yau threefolds with canonical singularities. The deformation space of such a Calabi-Yau threefold is no longer smooth, but the general principle is that the obstructions to deforming such a threefold are precisely the obstructions to deforming the singularities of the threefold. Secondly, these results are applied to smoothing singular Calabi-Yau threefolds with crepant resolutions. Any such Calabi-Yau threefold with isolated complete intersection singularities which are not ordinary double points is smoothable. A Calabi-Yau threefold with non-complete intersection isolated singularities is proved to be smoothable under much stronger hypotheses.