Conifold-transition connectivity of Calabi–Yau threefolds

Determine whether all complex threefolds with trivial canonical bundle can be connected by a sequence of conifold transitions, consisting of contractions of disjoint (-1,-1) curves followed by smoothings of ordinary double-point singularities.

Background

The paper places its study of non-Kähler solutions of the IIB system in the context of Reid’s fantasy: the proposed existence of an irreducible moduli space of algebraic Calabi–Yau manifolds with varying topology. Conifold transitions are a principal geometric mechanism for changing the topology of Calabi–Yau threefolds, but it is unresolved whether they suffice to connect all complex threefolds with trivial canonical bundle.

The metric constructions developed in the paper concern local models for conifold transitions, namely the deformed and resolved conifold. They provide non-compact cohomogeneity-one solutions of the IIB system, but do not resolve the global connectivity problem for all complex threefolds with trivial canonical bundle.