So, I was reading the famous Gopakumar Vafa paper, and they mention that $T^*S^3$ is a conifold. Why is this the case? I would naively expect $T^*S^3$ to be basically the same everywhere ($S^3$ is a maximally symmetric space, so I kind of expected its cotangent bundle to be nice as well). So where are the conical singularities coming from?


The conifold singularity is the quadric hypersurface singularity given in complex coordinates by


It is also known as the 3-fold ordinary double point. You can smooth this singularity by perturbing the equation:


(for some $\epsilon\neq 0$). This variety is smooth ("looks the same everywhere" in the sense that it's a complex manifold). It is diffeomorphic to the cotangent (or equivalently tangent) bundle of $S^3$: the zero section is just the real locus (assuming $\epsilon$ is real). You can write down a diffeomorphism explicitly, e.g.


So $TS^3$ is a smoothing ("the Milnor fibre") of the conifold singularity.

This singularity also admits a small resolution, which is the complex manifold given by the total space of the bundle $\mathcal O(-1)\oplus \mathcal O(-1)\to \mathbb{P}^1$. The process of degenerating the quadric to become singular and then resolving is often called a conifold transition.


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