一类Kähler锥的光滑性
On the smoothability of certain Kähler cones
摘要：
设$D$是一个Fano流形，对于某个Fano流形$Z$上的某秩$2$全纯向量丛$\mathcal{E}\longrightarrow Z$，它可以被实现为$\mathbb{P}(\mathcal{E})$。令$k\in\mathbb{N}$除$c_{1}(D)$。我们对可光滑的形式为$(\frac{1}{k}K_{D})^{\times}$的尺寸为$\leq4$的Kähler锥进行分类。因此，我们发现，任何不规则的卡-丘锥的尺寸$\leq 4$这种形式不承认一个平滑，离开$K_{\mathbb{P}^{2}_{(2)}}^{\times}$作为目前唯一已知的例子，一个平滑的不规则卡-丘锥在这些方面。
英文摘要：
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 Kähler 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.
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