标题： 具有孤立锥形奇点的卡拉比-丘流形 显示英文标题

标题： Calabi-Yau manifolds with isolated conical singularities

摘要： 设 $X$ 为具有典范奇点且典范丛平凡的复射影簇。 设 $L$ 为 $X$ 上的丰满线丛。 假设对偶组 $(X,L)$ 是一族光滑极化卡拉比-丘流形的平坦极限。 假设对于每个奇点 $x \in X$，存在一个 Kähler-Einstein Fano 流形 $Z$和一个正整数 $q$整除 $K_Z$，使得 $-\frac{1}{q}K_Z$非常丰富，并且使得芽 $(X,x)$在解析上局部同构于 $\frac{1}{q}K_{Z}$的零截面消解的顶点邻域。 我们证明了，在双全纯同构的意义下，唯一表示$2\pi c_1(L)$的弱Ricci平坦Kähler度量在$X$上在接近$x$时以多项式速率渐近于使用Calabi方法构造出的$\frac{1}{q}K_Z$上的自然Ricci平坦Kähler锥度量。特别地，如果$(X, \mathcal{O}(1))$是$\mathbb{P}^4$中的一个节点五次三维流形，我们的结果适用。这提供了首个已知的紧致Ricci平坦流形的例子，其具有非 orbifold 类型的孤立锥形奇点。 显示英文摘要

摘要： Let $X$ be a complex projective variety with only canonical singularities and with trivial canonical bundle. Let $L$ be an ample line bundle on $X$. Assume that the pair $(X,L)$ is the flat limit of a family of smooth polarized Calabi-Yau manifolds. Assume that for each singular point $x \in X$ there exist a Kahler-Einstein Fano manifold $Z$ and a positive integer $q$ dividing $K_Z$ such that $-\frac{1}{q}K_Z$ is very ample and such that the germ $(X,x)$ is locally analytically isomorphic to a neighborhood of the vertex of the blow-down of the zero section of $\frac{1}{q}K_{Z}$. We prove that up to biholomorphism, the unique weak Ricci-flat Kahler metric representing $2\pi c_1(L)$ on $X$ is asymptotic at a polynomial rate near $x$ to the natural Ricci-flat Kahler cone metric on $\frac{1}{q}K_Z$ constructed using the Calabi ansatz. In particular, our result applies if $(X, \mathcal{O}(1))$ is a nodal quintic threefold in $\mathbb{P}^4$. This provides the first known examples of compact Ricci-flat manifolds with non-orbifold isolated conical singularities.