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 Kähler-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 Kähler metric representing \(2\pi c_{1}(L)\) on \(X\) is asymptotic at a polynomial rate near \(x\) to the natural Ricci-flat Kähler 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 \(\mathbf {P}^{4}\). This provides the first known examples of compact Ricci-flat manifolds with non-orbifold isolated conical singularities.

中文翻译：

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具有孤立圆锥奇点的Calabi-Yau流形

令\（X \）是只有规范奇异性和平凡规范束的复杂射影变体。令\（L \）为\（X \）上足够的线束。假设\（（X，L）\）对是一族平滑极化的Calabi-Yau流形的平坦极限。假设对于每个奇点\（x \ in X \）存在一个Kähler-EinsteinFano流形\（Z \）和一个正整数\（q \）除以\（K_ {Z} \）使得\（- \ frac {1} {q} K_ {Z} \）非常大，这样细菌（（X，x）\）就局部解析同构到零截面的爆破顶点附近\（\ frac {1} {q} K_ {Z} \）。我们证明直到全同构，表示\（X \）上的\（2 \ pi c_ {1}（L）\）的唯一弱Ricci-flatKähler度量以接近自然Ricci-flatKähler锥度量\（x \）的多项式速率渐近在使用Calabi ansatz构造的\（\ frac {1} {q} K_ {Z} \）上。特别是，如果\（（X，\ mathcal {O}（1））\）是\（\ mathbf {P} ^ {4} \）中的节点五进制三倍，则我们的结果适用。这提供了具有非球面隔离的圆锥形奇点的紧凑Ricci-扁平歧管的第一个已知示例。