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Calabi-Yau manifolds with isolated conical singularities

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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 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.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Fordham University, Bronx, NY, 10458, USA

    Hans-Joachim Hein

  2. Department of Mathematics, Stony Brook University, Stony Brook, NY, 11790, USA

    Song Sun

Authors
  1. Hans-Joachim Hein
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  2. Song Sun
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Correspondence to Song Sun.

Additional information

H.-J. Hein is partially supported by NSF grant DMS-1514709.

S. Sun is partially supported by NSF grant DMS-1405832 and an Alfred P. Sloan Fellowship.

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Hein, HJ., Sun, S. Calabi-Yau manifolds with isolated conical singularities. Publ.math.IHES 126, 73–130 (2017). https://doi.org/10.1007/s10240-017-0092-1

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