Calabi-Yau manifolds with isolated conical singularities

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 ∈ X there exist a Kähler-Einstein Fano manifold Z and a positive integer q dividing K Z such that − 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 1 q K Z . We prove that up to biholomorphism, the unique weak Ricci-flat Kähler metric representing 2 π c 1 ( L ) on X is asymptotic at a polynomial rate near x to the natural Ricci-flat Kähler cone metric on 1 q K Z constructed using the Calabi ansatz. In particular, our result applies if ( X , O ( 1 ) ) is a nodal quintic threefold in P 4 . This provides the first known examples of compact Ricci-flat manifolds with non-orbifold isolated conical singularities.