On the smoothability of certain Kähler cones

Abstract

Let $D$ be a Fano manifold that may be realised as $\mathbb{P}(\mathcal{E})$ for some rank $2$ holomorphic vector bundle $\mathcal{E}⟶Z$ over some Fano manifold $Z$. Let $k\in \mathbb{N}$ divide $c_1(D)$. We classify those Kähler cones of dimension $\le 4$ of the form $(\frac{1}{k}{K}_D{)}^{\times }$ that are smoothable. As a consequence, we find that any irregular Calabi-Yau cone of dimension $\le 4$ of this form does not admit a smoothing, leaving $K_{{\mathbb{P}}_{(2)}^2}^{\times }$ as currently the only known example of a smoothable irregular Calabi-Yau cone in these dimensions.

4 pages

References in corpus (1)

• Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds