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.
Categories: math.AG