Conlon, Ronan J.; Hein, Hans-Joachim Classification of asymptotically conical Calabi-Yau manifolds. (English) Zbl 1548.14123 Duke Math. J. 173, No. 5, 947-1015 (2024). This paper is concerned with the construction and the classification of asymptotically conical Calabi-Yau metrics, which marks the completion of the authors’ program. The problem of complete non-compact Calabi-Yau metrics has a long history dating at least back to the seminal work of G. Tian and S. T. Yau [Invent. Math. 106, No. 1, 27–60 (1991; Zbl 0766.53053)]. In the last decade it has received a number of further impetus from Donaldson-Sun theory and the discovery of many new complete Calabi-Yau metrics, in which the authors played a major part. The general expectation is that under the assumption of maximal volume growth, so that Cheeger-Colding theory applies, there is a unique tangent cone at infinity, which is a possibly singular Calabi-Yau conical metric, and can be determined from algebraic geometric data; once the cone is fixed, the complete Calabi-Yau metric should be understood in terms of test configurations degenerating into the cone. The case of asymptotically conical metrics sits inside this big picture, as the special case where the tangent cone has a smooth link. This connects nicely with the celebrated uniqueness of tangent cone theorem by T. H. Colding and W. P. Minicozzi II [Invent. Math. 196, No. 3, 515–588 (2014; Zbl 1302.53048)] on Ricci flat manifolds, and the recent work of S. Sun and J. Zhang [Invent. Math. 233, No. 1, 461–494 (2023; Zbl 1519.14038)] sheds new light on the Riemannian geometric origin of asymptotic conicality, along with new progress on the algebro-geometric determination of the tangent cone.In the previous works of the authors’, they primarily focused on the case where the tangent cone corresponds to a quasi-regular Sasaki-Einstein metric, and in particular constructed complete Calabi-Yau metrics on the smoothings and the crepant resolutions of these cones. Once one has a version of the Tian-Yau existence theorem at one’s disposal, the construction question is mainly about finding a suitable diffeomorphism of the asymptotic region on the Calabi-Yau manifold with the tangent cone, such that one can put an asymptotically conical Kähler metric whose deviation from being Calabi-Yau decays sufficiently fast. The authors’ strategy depends on the compactification of the Calabi-Yau manifold by adding in a divisor at infinity, and the construction of the diffeomorphism comes from the exponential map around the divisor at infinity. In the quasi-regular case, there is a natural candidate divisor which is in general an orbifold coming from the circle quotient of the Sasaki-Einstein manifold. Now the challenge of the general case, which is tackled in this paper, is that the Reeb vector field may be irregular, and to get this compactification divisor, one needs to take approximation by quasi-regular Reeb vector fields, so that one obtains a sequence of different compactifications, and much technical work in this paper is involved in making the process uniform, including a generalization of some technical result of C. Li [Duke Math. J. 169, No. 8, 1397–1483 (2020; Zbl 1447.32018)] on the optimal rate of convergence for the complex structure.In the classification direction, the main task is that once the tangent cone at infinity is fixed, one needs to show that the asymptotically conical Calabi-Yau metric comes from a suitable test configuration. The strategy involves two parts: producing a test configuration from the asymptotically conical Calabi-Yau metric, and showing that the test configuration reproduces the original Calabi-Yau metric via the existence result.For the first part, one first needs to compactify the Calabi-Yau manifold by adding a divisor at infinity, and prove the ampleness of the divisor, and in particular the projectivity of the compactification. The choice of the divisor again depends on the quasi-regular approximation of the possibly irregular Reeb vector field, and the arguments involve some generalities on Stein spaces, plus some discussion of the normal bundle of the divisor at infinity. The asymptotically conical Calabi-Yau metric turns out to be the crepant resolution of some (possibly) singular Gorenstein affine variety with canonical singularities. Morally this means that resolution and smoothing are the only mechanisms of getting an asymptotically conical Calabi-Yau metric from the tangent cone. Once the compactification is in place, then one uses deformation of the normal cone to produce the required test configuration. For the second part, one needs to compare metric asymptotes and invoke the uniqueness theorem previously proved by the same authors [Duke Math. J. 162, No. 15, 2855–2902 (2013; Zbl 1283.53045)]. Here the issue of automorphism adds some technical complications.To apply the classification, one needs to understand the concrete source of the test configurations. The problem is roughly about first finding the versal deformation space of the cone, and then asking for an equivariant complex torus action. This is carried out in some cases including the Stenzel quadric cone, and a new proof of P. B. Kronheimer’s classification of ALE gravitational instantons [J. Differ. Geom. 29, No. 3, 665–683 (1989; Zbl 0671.53045)]. Reviewer: Yang Li (Cambridge) Cited in 1 Document MSC: 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) Keywords:asymptotically conical Calabi-Yau metrics; Euclidean volume growth; resolutions of tangent cones; test configurations; compactifications; ALE gravitational instantons; Tian-Yau construction Citations:Zbl 0766.53053; Zbl 1302.53048; Zbl 1519.14038; Zbl 1447.32018; Zbl 1283.53045; Zbl 0671.53045 × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] M. ABATE, F. BRACCI, and F. TOVENA, Embeddings of submanifolds and normal bundles, Adv. Math. 220 (2009), no. 2, 620-656. Digital Object Identifier: 10.1016/j.aim.2008.10.001 Google Scholar: Lookup Link MathSciNet: MR2466428 · Zbl 1161.32011 · doi:10.1016/j.aim.2008.10.001 [2] L. ALESSANDRINI and G. BASSANELLI, On the embedding of 1-convex manifolds with 1-dimensional exceptional set, Ann. Inst. Fourier (Grenoble) 51 (2001), no. 1, 99-108. MathSciNet: MR1821070 · Zbl 0966.32008 [3] K. ALTMANN, Minkowski sums and homogeneous deformations of toric varieties, Tohoku Math. J. (2) 47 (1995), no. 2, 151-184. Digital Object Identifier: 10.2748/tmj/1178225590 Google Scholar: Lookup Link MathSciNet: MR1329519 · Zbl 0842.14037 · doi:10.2748/tmj/1178225590 [4] K. ALTMANN, The versal deformation of an isolated toric Gorenstein singularity, Invent. Math. 128 (1997), no. 3, 443-479. Digital Object Identifier: 10.1007/s002220050148 Google Scholar: Lookup Link MathSciNet: MR1452429 · Zbl 0894.14025 · doi:10.1007/s002220050148 [5] K. ALTMANN, \( Toric \mathbb{Q} \)-Gorenstein singularities, preprint, arXiv:alg-geom/9403003v1. [6] M. T. ANDERSON, On the topology of complete manifolds of nonnegative Ricci curvature, Topology 29 (1990), no. 1, 41-55. Digital Object Identifier: 10.1016/0040-9383(90)90024-E Google Scholar: Lookup Link MathSciNet: MR1046624 · Zbl 0696.53027 · doi:10.1016/0040-9383(90)90024-E [7] M. T. ANDERSON, “A survey of Einstein metrics on 4-manifolds” in Handbook of Geometric Analysis, No. 3, Adv. Lect. Math. (ALM) 14, Int. Press, Somerville, 2010, 1-39. MathSciNet: MR2743446 · Zbl 1210.53050 [8] M. ARTIN, Versal deformations and algebraic stacks, Invent. Math. 27 (1974), 165-189. Digital Object Identifier: 10.1007/BF01390174 Google Scholar: Lookup Link MathSciNet: MR0399094 · Zbl 0317.14001 · doi:10.1007/BF01390174 [9] M. ARTIN, Lectures on Deformations of Singularities, Tata Institute of Fundamental Research, Bombay, 1976, https://archive.org/details/Michael_Artin___Lectures_on_Deformations_of_Singularities · Zbl 0395.14003 [10] W. L. BAILY, On the imbedding of V-manifolds in projective space, Amer. J. Math. 79 (1957), 403-430. Digital Object Identifier: 10.2307/2372689 Google Scholar: Lookup Link MathSciNet: MR0100104 · Zbl 0173.22706 · doi:10.2307/2372689 [11] S. BANDO, A. KASUE, and H. NAKAJIMA, On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth, Invent. Math. 97 (1989), no. 2, 313-349. Digital Object Identifier: 10.1007/BF01389045 Google Scholar: Lookup Link MathSciNet: MR1001844 · Zbl 0682.53045 · doi:10.1007/BF01389045 [12] S. BANDO and R. KOBAYASHI, Ricci-flat Kähler metrics on affine algebraic manifolds, II, Math. Ann. 287 (1990), no. 1, 175-180. Digital Object Identifier: 10.1007/BF01446884 Google Scholar: Lookup Link MathSciNet: MR1048287 · Zbl 0701.53083 · doi:10.1007/BF01446884 [13] D. BERENSTEIN, P. OUYANG, S. B. PINANSKY, and C. P. HERZOG, Supersymmetry breaking from a Calabi-Yau singularity, J. High Energy Phys. 2005, no. 9, art. ID 084. Digital Object Identifier: 10.1088/1126-6708/2005/09/084 Google Scholar: Lookup Link MathSciNet: MR2174138 · doi:10.1088/1126-6708/2005/09/084 [14] O. BIQUARD and T. DELCROIX, Ricci flat Kähler metrics on rank two complex symmetric spaces, J. Éc. polytech. Math. 6 (2019), 163-201. Digital Object Identifier: 10.5802/jep.91 Google Scholar: Lookup Link MathSciNet: MR3932737 · Zbl 1431.53076 · doi:10.5802/jep.91 [15] O. BIQUARD and P. GAUDUCHON, “Hyper-Kähler metrics on cotangent bundles of Hermitian symmetric spaces” in Geometry and Physics (Aarhus, 1995), Lect. Notes Pure Appl. Math. 184, Dekker, New York, 1997, 287-298. MathSciNet: MR1423175 · Zbl 0879.53051 [16] C. P. Boyer and K. Galicki, Sasakian Geometry, Oxford Math. Monogr., Oxford Univ. Press, Oxford, 2008. MathSciNet: MR2382957 · Zbl 1155.53002 [17] P. Candelas and X. C. de la Ossa, Comments on conifolds, Nuclear Phys. B 342 (1990), no. 1, 246-268. Digital Object Identifier: 10.1016/0550-3213(90)90577-Z Google Scholar: Lookup Link MathSciNet: MR1068113 · doi:10.1016/0550-3213(90)90577-Z [18] H. Cartan, “Quotient d’un espace analytique par un groupe d’automorphismes” in Algebraic Geometry and Topology, Princeton Univ. Press, Princeton, 1957, 90-102. MathSciNet: MR0084174 MathSciNet: MR84174 · Zbl 0084.07202 [19] J. CHEEGER, Degeneration of Riemannian metrics under Ricci curvature bounds, Fermi Lectures, Scuola Normale Superiore, Pisa, 2001. MathSciNet: MR2006642 · Zbl 1055.53024 [20] J. CHEEGER, Degeneration of Einstein metrics and metrics with special holonomy, Surv. Diff. Geom. 8 (2003), 29-73. Digital Object Identifier: 10.4310/SDG.2003.v8.n1.a2 Google Scholar: Lookup Link MathSciNet: MR2039985 · Zbl 1053.53028 · doi:10.4310/SDG.2003.v8.n1.a2 [21] J. Cheeger and T. H. Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. (2) 144 (1996), no. 1, 189-237. Digital Object Identifier: 10.2307/2118589 Google Scholar: Lookup Link MathSciNet: MR1405949 · Zbl 0865.53037 · doi:10.2307/2118589 [22] J. CHEEGER and A. NABER, Regularity of Einstein manifolds and the codimension 4 conjecture, Ann. of Math. (2) 182 (2015), no. 3, 1093-1165. Digital Object Identifier: 10.4007/annals.2015.182.3.5 Google Scholar: Lookup Link MathSciNet: MR3418535 · Zbl 1335.53057 · doi:10.4007/annals.2015.182.3.5 [23] J. CHEEGER and G. TIAN, On the cone structure at infinity of Ricci flat manifolds with Euclidean volume growth and quadratic curvature decay, Invent. Math. 118 (1994), no. 3, 493-571. Digital Object Identifier: 10.1007/BF01231543 Google Scholar: Lookup Link MathSciNet: MR1296356 · Zbl 0814.53034 · doi:10.1007/BF01231543 [24] S.-K. CHIU, Nonuniqueness of Calabi-Yau metrics with maximal volume growth, preprint, arXiv:2206.08210v1 [math.DG]. [25] S.-K. CHIU, Subquadratic harmonic functions on Calabi-Yau manifolds with Euclidean volume growth, to appear in Comm. Pure Appl. Math., preprint, arXiv:1905.12965v2 [math.DG]. [26] K. CHO, A. FUTAKI, and H. ONO, Uniqueness and examples of compact toric Sasaki-Einstein metrics, Comm. Math. Phys. 277 (2008), no. 2, 439-458. Digital Object Identifier: 10.1007/s00220-007-0374-4 Google Scholar: Lookup Link MathSciNet: MR2358291 · Zbl 1144.53058 · doi:10.1007/s00220-007-0374-4 [27] T. H. COLDING and W. P. MINICOZZI II, On uniqueness of tangent cones for Einstein manifolds, Invent. Math. 196 (2014), no. 3, 515-588. Digital Object Identifier: 10.1007/s00222-013-0474-z Google Scholar: Lookup Link MathSciNet: MR3211041 · Zbl 1302.53048 · doi:10.1007/s00222-013-0474-z [28] T. C. COLLINS and G. K. SZÉKELYHIDI, K-semistability for irregular Sasakian manifolds, J. Differential Geom. 109 (2018), no. 1, 81-109. Digital Object Identifier: 10.4310/jdg/1525399217 Google Scholar: Lookup Link MathSciNet: MR3798716 · Zbl 1403.53039 · doi:10.4310/jdg/1525399217 [29] T. C. COLLINS and G. K. SZÉKELYHIDI, Sasaki-Einstein metrics and K-stability, Geom. Topol. 23 (2019), no. 3, 1339-1413. Digital Object Identifier: 10.2140/gt.2019.23.1339 Google Scholar: Lookup Link MathSciNet: MR3956894 · Zbl 1432.32033 · doi:10.2140/gt.2019.23.1339 [30] T. C. COLLINS and V. TOSATTI, A singular Demailly-Păun theorem, C. R. Math. Acad. Sci. Paris 354 (2016), no. 1, 91-95. Digital Object Identifier: 10.1016/j.crma.2015.10.012 Google Scholar: Lookup Link MathSciNet: MR3439731 · Zbl 1344.32005 · doi:10.1016/j.crma.2015.10.012 [31] R. J. CONLON, A. DEGERATU, and F. ROCHON, Quasi-asymptotically conical Calabi-Yau manifolds, with appendix “More examples of Kähler-Einstein orbifolds admitting a crepant resolution” by R. J. Conlon, F. Rochon, and L. Sektnan, Geom. Topol. 23 (2019), no. 1, 29-100. Digital Object Identifier: 10.2140/gt.2019.23.29 Google Scholar: Lookup Link MathSciNet: MR3921316 · Zbl 1412.53102 · doi:10.2140/gt.2019.23.29 [32] R. J. Conlon and H.-J. Hein, Asymptotically conical Calabi-Yau manifolds, I, Duke Math. J. 162 (2013), no. 15, 2855-2902. Digital Object Identifier: 10.1215/00127094-2382452 Google Scholar: Lookup Link MathSciNet: MR3161306 · Zbl 1283.53045 · doi:10.1215/00127094-2382452 [33] R. J. CONLON and H.-J. HEIN, Asymptotically conical Calabi-Yau metrics on quasi-projective varieties, Geom. Funct. Anal. 25 (2015), no. 2, 517-552. Digital Object Identifier: 10.1007/s00039-015-0319-6 Google Scholar: Lookup Link MathSciNet: MR3334234 · Zbl 1333.32032 · doi:10.1007/s00039-015-0319-6 [34] R. J. CONLON and F. ROCHON, New examples of complete Calabi-Yau metrics on (( \mathbb{C}^n for\) n \geq 3\), Ann. Sci. Éc. Norm. Supér. (4) 54 (2021), no. 2, 259-303. Digital Object Identifier: 10.24033/asens.2459 Google Scholar: Lookup Link MathSciNet: MR4258163 · Zbl 1477.32046 · doi:10.24033/asens.2459 [35] D. A. Cox, J. B. Little, and H. K. Schenck, Toric Varieties, Grad. Stud. Math. 124, Amer. Math. Soc., Providence, 2011. Digital Object Identifier: 10.1090/gsm/124 Google Scholar: Lookup Link MathSciNet: MR2810322 · Zbl 1223.14001 · doi:10.1090/gsm/124 [36] A. H. DOAN, A counter-example to the equivariance structure on semi-universal deformation, J. Geom. Anal. 31 (2021), no. 4, 3698-3712. Digital Object Identifier: 10.1007/s12220-020-00411-4 Google Scholar: Lookup Link MathSciNet: MR4236540 · Zbl 1460.14029 · doi:10.1007/s12220-020-00411-4 [37] S. K. DONALDSON, Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002), no. 2, 289-349. MathSciNet: MR1988506 · Zbl 1074.53059 [38] S. K. DONALDSON and S. SUN, Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry, II, J. Differential Geom. 107 (2017), no. 2, 327-371. Digital Object Identifier: 10.4310/jdg/1506650422 Google Scholar: Lookup Link MathSciNet: MR3707646 · Zbl 1388.53074 · doi:10.4310/jdg/1506650422 [39] A. H. DURFEE, Fifteen characterizations of rational double points and simple critical points, Enseign. Math. (2) 25 (1979), no. 1-2, 131-163. MathSciNet: MR0543555 · Zbl 0418.14020 [40] R. Elkik, Solutions d’équations à coefficients dans un anneau hensélien, Ann. Sci. Éc. Norm. Supér. (4) 6 (1973), 553-603. MathSciNet: MR0345966 · Zbl 0327.14001 [41] C. I. EPSTEIN and G. M. HENKIN, Stability of embeddings for pseudoconcave surfaces and their boundaries, Acta Math. 185 (2000), no. 2, 161-237. Digital Object Identifier: 10.1007/BF02392810 Google Scholar: Lookup Link MathSciNet: MR1819994 · Zbl 0983.32035 · doi:10.1007/BF02392810 [42] M. FAULK, Some canonical metrics on Kähler orbifolds, Ph.D. dissertation, Columbia University, New York, 2019. [43] W. FULTON, Intersection Theory, 2nd ed., Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin, 1998. Digital Object Identifier: 10.1007/978-1-4612-1700-8 Google Scholar: Lookup Link MathSciNet: MR1644323 · doi:10.1007/978-1-4612-1700-8 [44] A. Futaki, H. Ono, and G. Wang, Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds, J. Differential Geom. 83 (2009), no. 3, 585-635. Digital Object Identifier: 10.4310/JDG/1264601036 Google Scholar: Lookup Link MathSciNet: MR2581358 · Zbl 1188.53042 · doi:10.4310/JDG/1264601036 [45] J. P. GAUNTLETT, D. MARTELLI, J. SPARKS, and D. WALDRAM, Sasaki-Einstein metrics on \(S^2 \times S^3\), Adv. Theor. Math. Phys. 8 (2004), no. 4, 711-734. Digital Object Identifier: 10.4310/ATMP.2004.v8.n4.a3 Google Scholar: Lookup Link MathSciNet: MR2141499 · Zbl 1136.53317 · doi:10.4310/ATMP.2004.v8.n4.a3 [46] R. Goto, Calabi-Yau structures and Einstein-Sasakian structures on crepant resolutions of isolated singularities, J. Math. Soc. Japan 64 (2012), no. 3, 1005-1052. Digital Object Identifier: 10.2969/jmsj/06431005 Google Scholar: Lookup Link MathSciNet: MR2965437 zbMATH: 1262.53041 · Zbl 1262.53041 · doi:10.2969/jmsj/06431005 [47] H. GRAUERT, Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962), 331-368. Digital Object Identifier: 10.1007/BF01441136 Google Scholar: Lookup Link MathSciNet: MR0137127 · Zbl 0173.33004 · doi:10.1007/BF01441136 [48] H. GRAUERT, Über die Deformation isolierter Singularitäten analytischer Mengen, Invent. Math. 15 (1972), 171-198. Digital Object Identifier: 10.1007/BF01404124 Google Scholar: Lookup Link MathSciNet: MR0293127 · Zbl 0237.32011 · doi:10.1007/BF01404124 [49] H. GRAUERT and O. RIEMENSCHNEIDER, Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen, Invent. Math. 11 (1970), 263-292. Digital Object Identifier: 10.1007/BF01403182 Google Scholar: Lookup Link MathSciNet: MR0302938 · Zbl 0202.07602 · doi:10.1007/BF01403182 [50] G.-M. Greuel, C. Lossen, and E. Shustin, Introduction to Singularities and Deformations, Springer Monogr. Math., Springer, Berlin, 2007. MathSciNet: MR2290112 · Zbl 1125.32013 [51] P. A. GRIFFITHS, The extension problem in complex analysis, II: Embeddings with positive normal bundle, Amer. J. Math. 88 (1966), 366-446. Digital Object Identifier: 10.2307/2373200 Google Scholar: Lookup Link MathSciNet: MR0206980 · Zbl 0147.07502 · doi:10.2307/2373200 [52] A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique, II: Étude globale élémentaire de quelques classes de morphismes, Publ. Math. Inst. Hautes Études Sci. 8 (1961). MathSciNet: MR0217084 [53] R. Hartshorne, Algebraic Geometry, Grad. Texts in Math. 52, Springer, New York, 1977. Digital Object Identifier: 10.1007/978-1-4757-3849-0 Google Scholar: Lookup Link MathSciNet: MR0463157 · Zbl 0367.14001 · doi:10.1007/978-1-4757-3849-0 [54] H. HAUSER, La construction de la déformation semi-universelle d’un germe de variété analytique complexe, Ann. Sci. Éc. Norm. Supér. (4) 18 (1985), no. 1, 1-56. MathSciNet: MR0803194 · Zbl 0583.32052 [55] H.-J. HEIN and C. LEBRUN, Mass in Kähler geometry, Comm. Math. Phys. 347 (2016), no. 1, 183-221. Digital Object Identifier: 10.1007/s00220-016-2661-4 Google Scholar: Lookup Link MathSciNet: MR3543182 · Zbl 1352.53060 · doi:10.1007/s00220-016-2661-4 [56] H.-J. HEIN, R. RĂSDEACONU, and I. ŞUVAINA, On the classification of ALE Kähler manifolds, Int. Math. Res. Not. IMRN 2021, no. 14, 10957-10980. Digital Object Identifier: 10.1093/imrn/rnz376 Google Scholar: Lookup Link MathSciNet: MR4285740 · Zbl 1497.53121 · doi:10.1093/imrn/rnz376 [57] H.-J. Hein and S. Sun, Calabi-Yau manifolds with isolated conical singularities, Publ. Math. Inst. Hautes Études Sci. 126 (2017), 73-130. Digital Object Identifier: 10.1007/s10240-017-0092-1 Google Scholar: Lookup Link MathSciNet: MR3735865 · Zbl 1397.32009 · doi:10.1007/s10240-017-0092-1 [58] H. HIRONAKA and H. ROSSI, On the equivalence of imbeddings of exceptional complex spaces, Math. Ann. 156 (1964), 313-333. Digital Object Identifier: 10.1007/BF01361027 Google Scholar: Lookup Link MathSciNet: MR0171784 · Zbl 0136.20801 · doi:10.1007/BF01361027 [59] N. O. ILTEN and R. VOLLMERT, Deformations of rational T-varieties, J. Algebraic Geom. 21 (2012), no. 3, 531-562. Digital Object Identifier: 10.1090/S1056-3911-2011-00585-7 Google Scholar: Lookup Link MathSciNet: MR2914803 · Zbl 1244.14044 · doi:10.1090/S1056-3911-2011-00585-7 [60] S. ISHII, Introduction to Singularities, Springer, Tokyo, 2014. Digital Object Identifier: 10.1007/978-4-431-55081-5 Google Scholar: Lookup Link MathSciNet: MR3288750 · Zbl 1308.14001 · doi:10.1007/978-4-431-55081-5 [61] W. JIANG and A. NABER, \( L^2\) curvature bounds on manifolds with bounded Ricci curvature, Ann. of Math. (2) 193 (2021), no. 1, 107-222. Digital Object Identifier: 10.4007/annals.2021.193.1.2 Google Scholar: Lookup Link MathSciNet: MR4199730 · Zbl 1461.53009 · doi:10.4007/annals.2021.193.1.2 [62] D. JOYCE, Asymptotically locally Euclidean metrics with holonomy \(\operatorname{SU}(m)\), Ann. Global Anal. Geom. 19 (2001), no. 1, 55-73. Digital Object Identifier: 10.1023/A:1006622430781 Google Scholar: Lookup Link MathSciNet: MR1824171 · Zbl 1009.53052 · doi:10.1023/A:1006622430781 [63] D. JOYCE, Quasi-ALE metrics with holonomy (( \operatorname{SU}(m) and \operatorname{Sp} \)(m)\), Ann. Global Anal. Geom. 19 (2001), no. 2, 103-132. Digital Object Identifier: 10.1023/A:1010778214851 Google Scholar: Lookup Link MathSciNet: MR1826397 · Zbl 0981.58019 · doi:10.1023/A:1010778214851 [64] S. KARIGIANNIS and J. D. LOTAY, Deformation theory of \(G_2 conifolds \), Comm. Anal. Geom. 28 (2020), no. 5, 1057-1210. Digital Object Identifier: 10.4310/CAG.2020.v28.n5.a1 Google Scholar: Lookup Link MathSciNet: MR4165315 · Zbl 1475.53055 · doi:10.4310/CAG.2020.v28.n5.a1 [65] A. KAS and M. SCHLESSINGER, On the versal deformation of a complex space with an isolated singularity, Math. Ann. 196 (1972), 23-29. Digital Object Identifier: 10.1007/BF01419428 Google Scholar: Lookup Link MathSciNet: MR0294701 · Zbl 0242.32014 · doi:10.1007/BF01419428 [66] J. KOLLÁR, Flops, Nagoya Math. J. 113 (1989), 15-36. Digital Object Identifier: 10.1017/S0027763000001240 Google Scholar: Lookup Link MathSciNet: MR0986434 · Zbl 0645.14004 · doi:10.1017/S0027763000001240 [67] J. KOLLÁR, “Flips, flops, minimal models, etc.” in Surveys in Differential Geometry (Cambridge, MA, 1990), Lehigh Univ., Bethlehem, PA, 1991, 113-199. MathSciNet: MR1144527 · Zbl 0755.14003 [68] J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge Tracts in Math. 134, Cambridge Univ. Press, Cambridge, 1998. Digital Object Identifier: 10.1017/CBO9780511662560 Google Scholar: Lookup Link MathSciNet: MR1658959 · Zbl 0926.14003 · doi:10.1017/CBO9780511662560 [69] P. B. KRONHEIMER, The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom. 29 (1989), no. 3, 665-683. MathSciNet: MR0992334 · Zbl 0671.53045 [70] P. B. KRONHEIMER, A Torelli-type theorem for gravitational instantons, J. Differential Geom. 29 (1989), no. 3, 685-697. Digital Object Identifier: 10.4310/jdg/1214443067 Google Scholar: Lookup Link MathSciNet: MR0992335 · Zbl 0671.53046 · doi:10.4310/jdg/1214443067 [71] K. LAMOTKE, Regular Solids and Isolated Singularities, Adv. Lect. Math. (ALM), Vieweg, Braunschweig, 1986. Digital Object Identifier: 10.1007/978-3-322-91767-6 Google Scholar: Lookup Link MathSciNet: MR0845275 · Zbl 0584.32014 · doi:10.1007/978-3-322-91767-6 [72] C. LI, On sharp rates and analytic compactifications of asymptotically conical Kähler metrics, Duke Math. J. 169 (2020), no. 8, 1397-1483. Digital Object Identifier: 10.1215/00127094-2019-0073 Google Scholar: Lookup Link MathSciNet: MR4101736 · Zbl 1447.32018 · doi:10.1215/00127094-2019-0073 [73] P. LI, Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature, Ann. of Math. (2) 124 (1986), no. 1, 1-21. Digital Object Identifier: 10.2307/1971385 Google Scholar: Lookup Link MathSciNet: MR0847950 · Zbl 0613.58032 · doi:10.2307/1971385 [74] Y. LI, A new complete Calabi-Yau metric on \(\mathbb{C}^3\), Invent. Math. 217 (2019), no. 1, 1-34. Digital Object Identifier: 10.1007/s00222-019-00861-w Google Scholar: Lookup Link MathSciNet: MR3958789 · Zbl 1422.53061 · doi:10.1007/s00222-019-00861-w [75] G. LIU, Compactification of certain Kähler manifolds with nonnegative Ricci curvature, Adv. Math. 382 (2021), no. 107652. Digital Object Identifier: 10.1016/j.aim.2021.107652 Google Scholar: Lookup Link MathSciNet: MR4224050 · Zbl 1466.53081 · doi:10.1016/j.aim.2021.107652 [76] X. MA and G. MARINESCU, Holomorphic Morse Inequalities and Bergman Kernels, Progr. Math. 254, Birkhäuser, Basel, 2007. Digital Object Identifier: 10.1007/978-3-7643-8115-8 Google Scholar: Lookup Link MathSciNet: MR2339952 · Zbl 1135.32001 · doi:10.1007/978-3-7643-8115-8 [77] D. MARTELLI, J. SPARKS, and S.-T. YAU, The geometric dual of a-maximisation for toric Sasaki-Einstein manifolds, Comm. Math. Phys. 268 (2006), no. 1, 39-65. Digital Object Identifier: 10.1007/s00220-006-0087-0 Google Scholar: Lookup Link MathSciNet: MR2249795 · Zbl 1190.53041 · doi:10.1007/s00220-006-0087-0 [78] D. MARTELLI, J. SPARKS, and S.-T. YAU, Sasaki-Einstein manifolds and volume minimisation, Comm. Math. Phys. 280 (2008), no. 3, 611-673. Digital Object Identifier: 10.1007/s00220-008-0479-4 Google Scholar: Lookup Link MathSciNet: MR2399609 · Zbl 1161.53029 · doi:10.1007/s00220-008-0479-4 [79] Y. NITTA and K. SEKIYA, Uniqueness of Sasaki-Einstein metrics, Tohoku Math. J. (2) 64 (2012), no. 3, 453-468. Digital Object Identifier: 10.2748/tmj/1347369373 Google Scholar: Lookup Link MathSciNet: MR2979292 · Zbl 1253.53047 · doi:10.2748/tmj/1347369373 [80] A. N. PARSHIN and I. R. SHAFAREVICH, eds., Algebraic Geometry V, Encyclopedia Math. Sci. 47, Springer, Berlin, 1999. · Zbl 0903.00014 [81] S. PINANSKY, Quantum deformations from toric geometry, J. High Energy Phys. (2006), no. 3, art. ID 055. Digital Object Identifier: 10.1088/1126-6708/2006/03/055 Google Scholar: Lookup Link MathSciNet: MR2222855 · Zbl 1226.81221 · doi:10.1088/1126-6708/2006/03/055 [82] H. C. PINKHAM, Deformations of algebraic varieties with \(G_m action \), Astérisque 20, Soc. Math. France, Paris, 1974. MathSciNet: MR0376672 · Zbl 0304.14006 [83] G. V. RAVINDRA and V. SRINIVAS, The Grothendieck-Lefschetz theorem for normal projective varieties, J. Algebraic Geom. 15 (2006), no. 3, 563-590. Digital Object Identifier: 10.1090/S1056-3911-05-00421-2 Google Scholar: Lookup Link MathSciNet: MR2219849 · Zbl 1123.14004 · doi:10.1090/S1056-3911-05-00421-2 [84] O. RIEMENSCHNEIDER, Characterizing Moišezon spaces by almost positive coherent analytic sheaves, Math. Z. 123 (1971), 263-284. Digital Object Identifier: 10.1007/BF01114795 Google Scholar: Lookup Link MathSciNet: MR0294714 · Zbl 0218.14006 · doi:10.1007/BF01114795 [85] D. S. RIM, Equivariant G-structure on versal deformations, Trans. Amer. Math. Soc. 257 (1980), no. 1, 217-226. Digital Object Identifier: 10.2307/1998132 Google Scholar: Lookup Link MathSciNet: MR0549162 · Zbl 0456.14004 · doi:10.2307/1998132 [86] J. ROSS and R. THOMAS, Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics, J. Differential Geom. 88 (2011), no. 1, 109-159. MathSciNet: MR2819757 · Zbl 1244.32013 [87] H. ROSSI, Vector fields on analytic spaces, Ann. of Math. (2) 78 (1963), no. 3, 455-467. Digital Object Identifier: 10.2307/1970536 Google Scholar: Lookup Link MathSciNet: MR0162973 · Zbl 0129.29701 · doi:10.2307/1970536 [88] M. SCHLESSINGER, Functors of Artin rings, Trans. Amer. Math. Soc. 130 (1968), no. 2, 208-222. Digital Object Identifier: 10.2307/1994967 Google Scholar: Lookup Link MathSciNet: MR0217093 · Zbl 0167.49503 · doi:10.2307/1994967 [89] M. SCHLESSINGER, Rigidity of quotient singularities, Invent. Math. 14 (1971), 17-26. Digital Object Identifier: 10.1007/BF01418741 Google Scholar: Lookup Link MathSciNet: MR0292830 · Zbl 0232.14005 · doi:10.1007/BF01418741 [90] G. SEGAL, Equivariant K-theory, Publ. Math. Inst. Hautes Études Sci. 34 (1968), 129-151. MathSciNet: MR0234452 · Zbl 0199.26202 [91] P. SLODOWY, Simple Singularities and Simple Algebraic Groups, Lecture Notes in Math. 815, Springer, Berlin, 1980. MathSciNet: MR0584445 · Zbl 0441.14002 [92] M. B. STENZEL, Ricci-flat metrics on the complexification of a compact rank one symmetric space, Manuscripta Math. 80 (1993), no. 2, 151-163. Digital Object Identifier: 10.1007/BF03026543 Google Scholar: Lookup Link MathSciNet: MR1233478 · Zbl 0811.53049 · doi:10.1007/BF03026543 [93] S. SUN and J. ZHANG, No semistability at infinity for Calabi-Yau metrics asymptotic to cones, Invent. Math. 233 (2023), no. 1, 461-494. Digital Object Identifier: 10.1007/s00222-023-01187-4 Google Scholar: Lookup Link MathSciNet: MR4602001 · Zbl 1519.14038 · doi:10.1007/s00222-023-01187-4 [94] I. ŞUVAINA, ALE Ricci-flat Kähler metrics and deformations of quotient surface singularities, Ann. Global Anal. Geom. 41 (2012), no. 1, 109-123. Digital Object Identifier: 10.1007/s10455-011-9273-1 Google Scholar: Lookup Link MathSciNet: MR2860399 · Zbl 1236.53057 · doi:10.1007/s10455-011-9273-1 [95] G. SZÉKELYHIDI, Degenerations of \(\mathbf{C}^n\) and Calabi-Yau metrics, Duke Math. J. 168 (2019), no. 14, 2651-2700. Digital Object Identifier: 10.1215/00127094-2019-0021 Google Scholar: Lookup Link MathSciNet: MR4012345 · Zbl 1432.32034 · doi:10.1215/00127094-2019-0021 [96] G. SZÉKELYHIDI, Uniqueness of some Calabi-Yau metrics on \(\mathbf{C}^n\), Geom. Funct. Anal. 30 (2020), no. 4, 1152-1182. Digital Object Identifier: 10.1007/s00039-020-00543-3 Google Scholar: Lookup Link MathSciNet: MR4153912 · Zbl 1468.14076 · doi:10.1007/s00039-020-00543-3 [97] T. Takahashi, Deformations of Sasakian structures and its application to the Brieskorn manifolds, Tohoku Math. J. (2) 30 (1978), no. 1, 37-43. Digital Object Identifier: 10.2748/tmj/1178230095 Google Scholar: Lookup Link MathSciNet: MR0486587 · Zbl 0392.53025 · doi:10.2748/tmj/1178230095 [98] G. TIAN, “Aspects of metric geometry of four manifolds” in Inspired by S. S. Chern, Nankai Tracts Math. 11, World Sci., Hackensack, 2006, 381-397. Digital Object Identifier: 10.1142/9789812772688_0016 Google Scholar: Lookup Link MathSciNet: MR2313343 · doi:10.1142/9789812772688_0016 [99] G. Tian and S.-T. Yau, Complete Kähler manifolds with zero Ricci curvature, II, Invent. Math. 106 (1991), no. 1, 27-60. Digital Object Identifier: 10.1007/BF01243902 Google Scholar: Lookup Link MathSciNet: MR1123371 · Zbl 0766.53053 · doi:10.1007/BF01243902 [100] C. van Coevering, Ricci-flat Kähler metrics on crepant resolutions of Kähler cones, Math. Ann. 347 (2010), no. 3, 581-611. Digital Object Identifier: 10.1007/s00208-009-0446-1 Google Scholar: Lookup Link MathSciNet: MR2640044 · Zbl 1195.53100 · doi:10.1007/s00208-009-0446-1 [101] C. van Coevering, Examples of asymptotically conical Ricci-flat Kähler manifolds, Math. Z. 267 (2011), no. 1-2, 465-496. Digital Object Identifier: 10.1007/s00209-009-0631-7 Google Scholar: Lookup Link MathSciNet: MR2772262 · Zbl 1211.53072 · doi:10.1007/s00209-009-0631-7 [102] J. A. VIACLOVSKY, “Critical metrics for Riemannian curvature functionals” in Geometric Analysis, IAS/Park City Math. Ser. 22, Amer. Math. Soc., Providence, 2016, 197-274. Digital Object Identifier: 10.1090/pcms/022/05 Google Scholar: Lookup Link MathSciNet: MR3524218 · Zbl 1360.53005 · doi:10.1090/pcms/022/05 [103] E. P. WRIGHT, Quotients of gravitational instantons, Ann. Global Anal. Geom. 41 (2012), no. 1, 91-108. Digital Object Identifier: 10.1007/s10455-011-9272-2 Google Scholar: Lookup Link MathSciNet: MR2860398 · Zbl 1236.53060 · doi:10.1007/s10455-011-9272-2 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.