Asymptotically Calabi metrics and weak Fano manifolds

Hein, Hans-Joachim¹ ; Sun, Song² ; Viaclovsky, Jeffrey³ ; Zhang, Ruobing⁴

¹Mathematisches Institut, Universität Münster, Münster, Germany
²Institute for Advanced Study in Mathematics, Zhejiang University, Hangzhou, China
³Department of Mathematics, University of California, Irvine, Irvine, CA, United States
⁴Department of Mathematics, University of California, San Diego, CA, United States

Geometry & topology, Tome 29 (2025) no. 5, pp. 2547-2569

Résumé

We show that any asymptotically Calabi manifold which is Calabi–Yau can be compactified complex analytically to a weak Fano manifold. Furthermore, the Calabi–Yau structure arises from a generalized Tian–Yau construction on the compactification, and we prove a strong uniqueness theorem. We also give an application of this result to the surface case.

Détail

DOI : 10.2140/gt.2025.29.2547

Keywords: Calabi–Yau, weak Fano manifold, complex compactification

Bibliographie

[1] C Birkar, Singularities of linear systems and boundedness of Fano varieties, Ann. of Math. 193 (2021) 347 | DOI

[2] G Chen, X Chen, Gravitational instantons with faster than quadratic curvature decay, II, J. Reine Angew. Math. 756 (2019) 259 | DOI

[3] G Chen, X Chen, Gravitational instantons with faster than quadratic curvature decay, I, Acta Math. 227 (2021) 263 | DOI

[4] G Chen, X Chen, Gravitational instantons with faster than quadratic curvature decay, III, Math. Ann. 380 (2021) 687 | DOI

[5] G Chen, J Viaclovsky, Gravitational instantons with quadratic volume growth, J. Lond. Math. Soc. 109 (2024) | DOI

[6] Y Chen, Calabi–Yau metrics of Calabi type with polynomial rate of convergence, preprint (2024)

[7] C Van Coevering, A construction of complete Ricci-flat Kähler manifolds, preprint (2008)

[8] T C Collins, A Jacob, Y S Lin, Special Lagrangian submanifolds of log Calabi–Yau manifolds, Duke Math. J. 170 (2021) 1291 | DOI

[9] T C Collins, V Tosatti, A singular Demailly–Păun theorem, C. R. Math. Acad. Sci. Paris 354 (2016) 91 | DOI

[10] R J Conlon, H J Hein, Classification of asymptotically conical Calabi–Yau manifolds, Duke Math. J. 173 (2024) 947 | DOI

[11] J P Demailly, Complex analytic and differential geometry, book manuscript (2012)

[12] I V Dolgachev, Classical algebraic geometry: a modern view, Cambridge Univ. Press (2012) | DOI

[13] H Grauert, Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962) 331 | DOI

[14] H Grauert, O Riemenschneider, Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen, Invent. Math. 11 (1970) 263 | DOI

[15] B Harbourne, W E Lang, Multiple fibers on rational elliptic surfaces, Trans. Amer. Math. Soc. 307 (1988) 205 | DOI

[16] M Haskins, H J Hein, J Nordström, Asymptotically cylindrical Calabi–Yau manifolds, J. Differential Geom. 101 (2015) 213

[17] H J Hein, Weighted Sobolev inequalities under lower Ricci curvature bounds, Proc. Amer. Math. Soc. 139 (2011) 2943 | DOI

[18] H J Hein, Gravitational instantons from rational elliptic surfaces, J. Amer. Math. Soc. 25 (2012) 355 | DOI

[19] H J Hein, S Sun, J Viaclovsky, R Zhang, Nilpotent structures and collapsing Ricci-flat metrics on the K3 surface, J. Amer. Math. Soc. 35 (2022) 123 | DOI

[20] C D Hill, M Taylor, Integrability of rough almost complex structures, J. Geom. Anal. 13 (2003) 163 | DOI

[21] L Hörmander, An introduction to complex analysis in several variables, 7, North-Holland (1990)

[22] Y Kawamata, Pluricanonical systems on minimal algebraic varieties, Invent. Math. 79 (1985) 567 | DOI

[23] J Kollár, Y Miyaoka, S Mori, H Takagi, Boundedness of canonical Q-Fano 3-folds, Proc. Japan Acad. Ser. A Math. Sci. 76 (2000) 73

[24] J Kollár, S Mori, Birational geometry of algebraic varieties, 134, Cambridge Univ. Press (1998) | DOI

[25] P B Kronheimer, A Torelli-type theorem for gravitational instantons, J. Differential Geom. 29 (1989) 685

[26] C Li, On sharp rates and analytic compactifications of asymptotically conical Kähler metrics, Duke Math. J. 169 (2020) 1397 | DOI

[27] V Minerbe, Rigidity for multi-Taub-NUT metrics, J. Reine Angew. Math. 656 (2011) 47 | DOI

[28] M V Nori, Zariski’s conjecture and related problems, Ann. Sci. École Norm. Sup. 16 (1983) 305 | DOI

[29] T Ohsawa, Vanishing theorems on complete Kähler manifolds, Publ. Res. Inst. Math. Sci. 20 (1984) 21 | DOI

[30] I Shimada, Remarks on fundamental groups of complements of divisors on algebraic varieties, Kodai Math. J. 17 (1994) 311 | DOI

[31] S Sun, R Zhang, Complex structure degenerations and collapsing of Calabi–Yau metrics, preprint (2019)

[32] S Sun, R Zhang, A Liouville theorem on asymptotically Calabi spaces, Calc. Var. Partial Differential Equations 60 (2021) 103 | DOI

[33] S Sun, R Zhang, Collapsing geometry of hyperkähler 4-manifolds and applications, Acta Math. 232 (2024) 325 | DOI

[34] S Takayama, Simple connectedness of weak Fano varieties, J. Algebraic Geom. 9 (2000) 403

[35] G Tian, S T Yau, Complete Kähler manifolds with zero Ricci curvature, I, J. Amer. Math. Soc. 3 (1990) 579 | DOI

[36] C Voisin, Coniveau 2 complete intersections and effective cones, Geom. Funct. Anal. 19 (2010) 1494 | DOI

[37] S T Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation, I, Comm. Pure Appl. Math. 31 (1978) 339 | DOI

[38] S T Yau, The role of partial differential equations in differential geometry, from: "Proceedings of the International Congress of Mathematicians, I" (editor O Lehto), Acad. Sci. Fennica (1980) 237