Calabi-Yau manifolds with isolated conical singularities
Publications Mathématiques de l'IHÉS, Tome 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
@article{PMIHES_2017__126__73_0,
     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://archive.numdam.org/articles/10.1007/s10240-017-0092-1/}
}
TY  - JOUR
AU  - Hein, Hans-Joachim
AU  - Sun, Song
TI  - Calabi-Yau manifolds with isolated conical singularities
JO  - Publications Mathématiques de l'IHÉS
PY  - 2017
SP  - 73
EP  - 130
VL  - 126
PB  - Springer Berlin Heidelberg
PP  - Berlin/Heidelberg
UR  - http://archive.numdam.org/articles/10.1007/s10240-017-0092-1/
DO  - 10.1007/s10240-017-0092-1
LA  - en
ID  - PMIHES_2017__126__73_0
ER  - 
%0 Journal Article
%A Hein, Hans-Joachim
%A Sun, Song
%T Calabi-Yau manifolds with isolated conical singularities
%J Publications Mathématiques de l'IHÉS
%D 2017
%P 73-130
%V 126
%I Springer Berlin Heidelberg
%C Berlin/Heidelberg
%U http://archive.numdam.org/articles/10.1007/s10240-017-0092-1/
%R 10.1007/s10240-017-0092-1
%G en
%F PMIHES_2017__126__73_0
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. http://archive.numdam.org/articles/10.1007/s10240-017-0092-1/

[1.] Abraham, R.; Marsden, J.; Ratiu, T. Manifolds, Tensor Analysis, and Applications (1988) | DOI | Zbl

[2.] Ache, A.; Viaclovsky, J. Obstruction-flat asymptotically locally Euclidean metrics, Geom. Funct. Anal., Volume 22 (2012), pp. 832-877 | DOI | MR | Zbl

[3.] Andreotti, A.; Grauert, H. Théorèmes de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. Fr., Volume 90 (1962), pp. 193-259 | DOI | Numdam | Zbl

[4.] Arezzo, C.; Spotti, C. On cscK resolutions of conically singular cscK varieties, J. Funct. Anal., Volume 271 (2016), pp. 474-494 | DOI | MR | Zbl

[5.] Ballmann, W. Lectures on Kähler Manifolds (2006) | DOI | Zbl

[6.] Bando, S.; Mabuchi, T. Uniqueness of Einstein Kähler metrics modulo connected group actions, Algebraic Geometry (1987), pp. 11-40

[7.] Bănică, C. Le lieu réduit et le lieu normal d’un morphisme, Romanian-Finnish Seminar on Complex Analysis (1979), pp. 389-398 | DOI

[8.] Behrndt, T. On the Cauchy problem for the heat equation on Riemannian manifolds with conical singularities, Q. J. Math., Volume 64 (2013), pp. 981-1007 | DOI | MR | Zbl

[9.] Berline, N.; Getzler, E.; Vergne, M. Heat Kernels and Dirac Operators (2004) | Zbl

[10.] Berman, R. K-Polystability of Q-Fano varieties admitting Kähler-Einstein metrics, Invent. Math., Volume 203 (2016), pp. 973-1025 | DOI | MR | Zbl

[11.] Biquard, O.; Rollin, Y. Smoothing singular constant scalar curvature Kähler surfaces and minimal Lagrangians, Adv. Math., Volume 285 (2015), pp. 980-1024 | DOI | MR | Zbl

[12.] Calabi, E. Métriques kählériennes et fibrés holomorphes, Ann. Sci. Éc. Norm. Supér. (4), Volume 12 (1979), pp. 269-294 | DOI | Numdam | MR | Zbl

[13.] Candelas, P.; de la Ossa, X. Comments on conifolds, Nucl. Phys. B, Volume 342 (1990), pp. 246-268 | DOI | MR

[14.] Y-M. Chan, Calabi-Yau and Special Lagrangian 3-folds with conical singularities and their desingularizations, D.Phil. thesis, University of Oxford, 2005, https://people.maths.ox.ac.uk/joyce/theses/theses.html.

[15.] Cheeger, J. Spectral geometry of singular Riemannian spaces, J. Differ. Geom., Volume 18 (1983), pp. 575-657 | DOI | MR | Zbl

[16.] Cheeger, J.; Tian, G. On the cone structure at infinity of Ricci flat manifolds with Euclidean volume growth and quadratic curvature decay, Invent. Math., Volume 118 (1994), pp. 493-571 | DOI | MR | Zbl

[17.] Colding, T.; Minicozzi, W. On uniqueness of tangent cones for Einstein manifolds, Invent. Math., Volume 196 (2014), pp. 515-588 | DOI | MR | Zbl

[18.] T. Collins and G. Székelyhidi, K-Semistability for irregular Sasakian manifolds, J. Differ. Geom., preprint, | arXiv

[19.] Conlon, R.; Hein, H-J. Asymptotically conical Calabi-Yau manifolds, I, Duke Math. J., Volume 162 (2013), pp. 2855-2900 | DOI | MR | Zbl

[20.] R. Conlon and H-J. Hein, Asymptotically conical Calabi-Yau manifolds, III, preprint, | arXiv

[21.] Demailly, J-P.; Pali, N. Degenerate complex Monge-Ampère equations over compact Kähler manifolds, Int. J. Math., Volume 21 (2010), pp. 357-405 | DOI | Zbl

[22.] Donaldson, S. Kähler-Einstein Metrics and Algebraic Structures on Limit Spaces (2016), pp. 85-94 | Zbl

[23.] Donaldson, S.; Sun, S. Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry, Acta Math., Volume 213 (2014), pp. 63-106 | DOI | MR | Zbl

[24.] S. Donaldson and S. Sun, Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry, II, J. Differ. Geom., preprint, | arXiv

[25.] Eyssidieux, P.; Guedj, V.; Zeriahi, A. Singular Kähler-Einstein metrics, J. Am. Math. Soc., Volume 22 (2009), pp. 607-639 | DOI | Zbl

[26.] K. Fujita, Optimal bounds for the volumes of Kähler-Einstein Fano manifolds, Am. J. Math., preprint, | arXiv

[27.] Grigoryan, A.; Saloff-Coste, L. Stability results for Harnack inequalities, Ann. Inst. Fourier (Grenoble), Volume 55 (2005), pp. 825-890 | DOI | Numdam | MR | Zbl

[28.] Gubser, S. Einstein manifolds and conformal field theories, Phys. Rev. D (3), Volume 59 (1999) (8 pp.) | DOI | MR

[29.] H-J. Hein and A. Naber, Isolated Einstein singularities with singular tangent cones, in preparation.

[30.] Hörmander, L. L2 estimates and existence theorems for the ¯ operator, Acta Math., Volume 113 (1965), pp. 89-152 | DOI | MR | Zbl

[31.] Jonsson, M.; Mustaţă, M. Valuations and asymptotic invariants for sequences of ideals, Ann. Inst. Fourier (Grenoble), Volume 62 (2012), pp. 2145-2209 | DOI | Numdam | MR | Zbl

[32.] Joyce, D. Special Lagrangian submanifolds with isolated conical singularities, I, Ann. Glob. Anal. Geom., Volume 25 (2004), pp. 201-251 | DOI | MR | Zbl

[33.] Joyce, D. Special Lagrangian submanifolds with isolated conical singularities, III, Ann. Glob. Anal. Geom., Volume 26 (2005), pp. 1-58 | DOI | Zbl

[34.] C. Li, Minimizing normalized volumes of valuations, preprint, | arXiv

[35.] C. Li and Y-C. Liu, Kähler-Einstein metrics and volume minimization, preprint, | arXiv

[36.] C. Li and C. Xu, Stability of valuations and Kollár components, preprint, | arXiv

[37.] Martelli, D.; Sparks, J.; Yau, S-T. Sasaki-Einstein manifolds and volume minimisation, Commun. Math. Phys., Volume 280 (2007), pp. 611-673 | DOI | MR | Zbl

[38.] Mazzeo, R. Elliptic theory of edge operators, I, Commun. Partial Differ. Equ., Volume 16 (1991), pp. 1615-1664 | DOI | MR | Zbl

[39.] Morrey, C. Multiple Integrals in the Calculus of Variations (2008) | DOI | Zbl

[40.] Pacini, T. Desingularizing isolated conical singularities, Commun. Anal. Geom., Volume 21 (2013), pp. 105-170 | DOI | Zbl

[41.] Rong, X-C.; Zhang, Y-G. Continuity of extremal transitions and flops for Calabi-Yau manifolds, J. Differ. Geom., Volume 89 (2011), pp. 233-269 | DOI | MR | Zbl

[42.] Shustin, E.; Tyomkin, I. Versal deformation of algebraic hypersurfaces with isolated singularities, Math. Ann., Volume 313 (1999), pp. 297-314 | DOI | MR | Zbl

[43.] Simon, L. Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. Math. (2), Volume 118 (1983), pp. 525-571 | DOI | MR | Zbl

[44.] Smith, I.; Thomas, R. Symplectic surgeries from singularities, Turk. J. Math., Volume 27 (2003), pp. 231-250 | MR | Zbl

[45.] Song, J. On a conjecture of Candelas and de la Ossa, Commun. Math. Phys., Volume 334 (2015), pp. 697-717 | DOI | MR | Zbl

[46.] Spotti, C. Deformations of nodal Kähler-Einstein del Pezzo surfaces with discrete automorphism groups, J. Lond. Math. Soc., Volume 89 (2014), pp. 539-558 | DOI | MR | Zbl

[47.] Spotti, C.; Sun, S.; Yao, C-J. Existence and deformations of Kähler-Einstein metrics on smoothable Q-Fano varieties, Duke Math. J., Volume 165 (2016), pp. 3043-3083 | DOI | MR | Zbl

[48.] Stenzel, M. Ricci-flat metrics on the complexification of a compact rank one symmetric space, Manuscr. Math., Volume 80 (1993), pp. 151-163 | DOI | MR | Zbl

[49.] Tosatti, V. Limits of Calabi-Yau metrics when the Kähler class degenerates, J. Eur. Math. Soc., Volume 11 (2009), pp. 755-776 | DOI | MR | Zbl

[50.] Uhlenbeck, K. Removable singularities in Yang-Mills fields, Commun. Math. Phys., Volume 83 (1982), pp. 11-29 | DOI | MR | Zbl

[51.] C. van Coevering, A construction of complete Ricci-flat Kähler manifolds, preprint, | arXiv

[52.] B. Vertman, Ricci flow on singular manifolds, preprint, | arXiv

[53.] Wang, Y. On the Kähler-Ricci flows near the Mukai-Umemura 3-fold, Int. Math. Res. Not., Volume 2016 (2015), pp. 2145-2156 | DOI

[54.] Wehler, J. Deformation of complete intersections with singularities, Math. Z., Volume 179 (1982), pp. 473-491 | DOI | MR | Zbl

[55.] Yau, S-T. On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I, Commun. Pure Appl. Math., Volume 31 (1978), pp. 339-411 | DOI | Zbl

Cité par Sources :