An introduction to conifold transitions

Abstract

These lecture notes introduce conifold transitions between complex threefolds with trivial canonical bundle from the differential geometric point of view, and with a particular view towards aspects of mathematical physics and string theory. The lecture notes are aimed at beginning graduate students and non-experts, emphasizing explicit calculations and examples. After a brief introduction in Section 1, we recall some basic facts about Calabi-Yau manifolds in Section 2. Section 3 studies the conifold as a Calabi-Yau manifold with singularities, and introduces the local model for a conifold transition. Section 4 discusses global conifold transitions, and recalls the famous result of Friedman concerning the existence of smoothings for nodal Calabi-Yau threefolds. We give a differential geometric proof of the necessity part of Friedman's theorem. Section 5 discusses Reid's fantasy, and the web of Calabi-Yau threefolds. Section 6 discusses metric aspects of the local conifold transition, constructing explicit asymptotically conical Calabi-Yau metrics on the small resolution and the smoothing. Section 7 discusses the metric aspects of global conifold transitions, with a particular emphasis on the heterotic string.