Calabi-Yau manifolds with isolated conical singularities
Publications Mathématiques de l'IHÉS, Volume 126 (2017), pp. 73-130.

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 xX there exist a Kähler-Einstein Fano manifold Z and a positive integer q dividing KZ such that 1qKZ 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 1qKZ. We prove that up to biholomorphism, the unique weak Ricci-flat Kähler metric representing 2πc1(L) on X is asymptotic at a polynomial rate near x to the natural Ricci-flat Kähler cone metric on 1qKZ constructed using the Calabi ansatz. In particular, our result applies if (X,O(1)) is a nodal quintic threefold in P4. This provides the first known examples of compact Ricci-flat manifolds with non-orbifold isolated conical singularities.

DOI: 10.1007/s10240-017-0092-1
Hein, Hans-Joachim 1; Sun, Song 2

1 Department of Mathematics, Fordham University 10458 Bronx NY USA
2 Department of Mathematics, Stony Brook University 11790 Stony Brook NY USA
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Hein, Hans-Joachim; Sun, Song. Calabi-Yau manifolds with isolated conical singularities. Publications Mathématiques de l'IHÉS, Volume 126 (2017), pp. 73-130. doi : 10.1007/s10240-017-0092-1. http://archive.numdam.org/articles/10.1007/s10240-017-0092-1/

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