Degenerations of exotic Calabi-Yau metrics through Atiyah's flop

Abstract

We construct new families of complete Calabi-Yau metrics with maximal volume growth on the small resolutions of the conifold $\mathcal{Z}=\{z_1^2+z_2^2+z_3^2+z_4^2=0\}\subset {\mathbb{C}}^4$. These metrics have tangent cone $\mathbb{C}\times ({\mathbb{C}}^2/{\mathbb{Z}}_2)$ at infinity and are parametrised by their Kähler class. As the Kähler class degenerates, the metrics converge in the pointed Gromov-Hausdorff sense to a Calabi-Yau metric on $\mathcal{Z}$ with an isolated conical singularity modelled on the Stenzel metric at the ordinary double point and tangent cone at infinity $\mathbb{C}\times ({\mathbb{C}}^2/{\mathbb{Z}}_2)$, thereby providing a new metric realisation of the Atiyah flop.

65 pages, 1 appendix. Comments are welcome!