Calabi-Yau threefolds across quadratic singularities
Sébastien Picard
TL;DR
This work surveys Calabi–Yau threefolds, focusing on how conifold transitions—via contracting disjoint $(-1,-1)$-curves to ordinary double points and subsequent smoothing—connect different CY3 geometries and even yield non–Kähler limits. It develops a two-pronged geometrization: a local model where Calabi–Yau cone metrics interpolate between small resolutions and smoothings, and a global theory describing how Kähler Ricci-flat metrics deform and converge through these transitions, as well as non-Kähler generalizations of the Strominger system that persist across the transition. Key results include the local Calabi–Yau cone structure around conifold points, the Fried Friedman–Kawamata–Ran–Tian smoothing criterion, and the Ruan–Zhang–Rong–Zhang–Song convergence theorems illustrating metric continuity in a topologically discontinuous process. The work also discusses alternate geometric setups (balanced metrics, Type IIB flux, and symplectic viewpoints) and the broader “web” of complex threefolds connected by conifold transitions, highlighting how non-Kähler limits still retain structural CY features and moduli constraints with potential physical significance in string theory.
Abstract
These are lecture notes on non-Kähler complex threefolds presented at the MATRIX program ``The geometry of moduli spaces in string theory''. We review some basics of Calabi-Yau geometry in Section 1, describe topological features of the conifold transition in Section 2, and survey recent developments on the geometrization of conifold transitions in Section 3.