Calabi-Yau manifolds with isolated conical singularities

Hein, Hans-Joachim; Sun, Song

doi:10.1007/s10240-017-0092-1

Geodesic

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Calabi-Yau manifolds with isolated conical singularities

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

Publications Mathématiques de l'IHÉS, Tome 126 (2017), pp. 73-130

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Résumé

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 $2 π c_{1} (L)$ 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 $(X, O (1))$ 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.

MR Zbl

DOI : 10.1007/s10240-017-0092-1

Affiliations des auteurs :

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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     author = {Hein, Hans-Joachim and Sun, Song},
     title = {Calabi-Yau manifolds with isolated conical singularities},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {73--130},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {126},
     year = {2017},
     doi = {10.1007/s10240-017-0092-1},
     mrnumber = {3735865},
     zbl = {1397.32009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10240-017-0092-1/}
}

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Hein, Hans-Joachim; Sun, Song. Calabi-Yau manifolds with isolated conical singularities. Publications Mathématiques de l'IHÉS, Tome 126 (2017), pp. 73-130. doi: 10.1007/s10240-017-0092-1

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