On the smoothability of certain K\"ahler cones

TL;DR

This paper classifies smoothable K"ahler cones of certain dimensions arising from Fano manifolds and shows that most irregular Calabi-Yau cones of these types are not smoothable, except for a unique known example.

Contribution

It provides a classification of smoothable K"ahler cones of dimension up to 4 from specific Fano manifolds and identifies the only known smoothable irregular Calabi-Yau cone in these dimensions.

Findings

Classified smoothable K"ahler cones of dimension ≤ 4 from Fano manifolds.

Proved most irregular Calabi-Yau cones of this form are not smoothable.

Identified $K_{P^{2}_{(2)}}^{ imes}$ as the only known smoothable irregular Calabi-Yau cone in these dimensions.

Abstract

Let $D$ be a Fano manifold that may be realised as $\mathbb{P}(\mathcal{E})$ for some rank $2$ holomorphic vector bundle $\mathcal{E}\longrightarrow Z$ over some Fano manifold $Z$. Let $k\in\mathbb{N}$ divide $c_{1}(D)$. We classify those K\"ahler cones of dimension $\leq4$ 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 $\leq 4$ 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.