Calabi-Yau manifolds with isolated conical singularities

Abstract

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 $2\ddot{I}€c_1(L)$ 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.

41 pages, added a short appendix on special Lagrangian vanishing cycles