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## 4 ASYMPTOTICALLY CONICAL CALABI-YAU MANIFOLDS, III

Citations: | 8 - 3 self |

### Citations

2100 | Principles of algebraic geometry - Griffiths, Harris - 1994 |

1339 | Intersection Theory - Fulton - 1984 |

466 |
Compact manifolds with special holonomy
- Joyce
- 2000
(Show Context)
Citation Context ...lds, then at least heuristically, these are precisely the ones that correspond to noncollapsed isolated singularities. Besides [17], the foundational paper in this area is Tian-Yau [53], but see also =-=[27, 36, 56]-=-. Remark 1.1. Ricci-flat Kähler manifolds of Euclidean volume growth whose tangent cones are not smooth, and in fact not even products of smooth cones, exist as well: Joyce’s QALE spaces [36] are des... |

322 | Birational geometry of algebraic varieties, - Kollár, Mori - 1998 |

296 |
de la Ossa
- Candelas, C
- 1990
(Show Context)
Citation Context ... then C is the 3-fold ordinary double point, and either M = T ∗S3, the smoothing of C, with Stenzel’s metric [30], or else M = OP1(−1) ⊕2, the small resolution of C, with Candelas-de la Ossa’s metric =-=[4]-=-. (iv) If D is a quadric in Pn with n > 3 and k = n−1, then C is the n-fold ordinary double point, and M = T ∗Sn, the smoothing of C, with Stenzel’s metric [30]. 1Standard results in deformation theor... |

269 |
The construction of ALE spaces as hyper-Kähler quotients
- Kronheimer
- 1989
(Show Context)
Citation Context ...der to be able to use that the algebraic deformations of the algebraic surface singularity C2/Γ can be classified. Corollary C (Kronheimer [18]). Every AC Calabi-Yau surface is a Kronheimer ALE space =-=[17]-=-, i.e. a crepant resolution of a deformation of a Kleinian singularity C2/Γ with Γ as above. Proof. Consider the normal projective surface Y of Theorem A(ii), which contains D as an ample suborbifold ... |

222 | Functors of Artin rings
- Schlessinger
- 1968
(Show Context)
Citation Context ...rm the pair (Y,D) of Theorem A(ii) to its normal cone. The affine cone in (i) is rigid, whereas the ones in (ii), (iii), (iv) have exactly one deformation, which is smooth. (For (i) this follows from =-=[27]-=-, for (i), (ii), (iii) from [1], and for (iii), (iv) from [13].) Alternatively, we can apply the classification theory of log-Fano varieties [28, Definition 2.1.1] to Y : in (i) and (ii), (Y,D) is a d... |

192 | Characteristic classes and homogeneous spaces - Borel, Hirzebruch - 1958 |

186 |
Simple Singularities and Simple Algebraic Groups,
- Slodowy
- 1980
(Show Context)
Citation Context ...on in order to be able to pull Y back from the versal deformation. We can undo this shrinking here by using the existence of a C∗-equivariant map to the versal C∗-deformation in the analytic category =-=[29]-=-. A very similar point appears in [18, (2.5)]. Asymptotically conical Calabi-Yau manifolds, III 3 Proof. We again deform the pair (Y,D) of Theorem A(ii) to its normal cone. The affine cone in (i) is r... |

158 | Symplectic submanifolds and almost-complex geometry, - Donaldson - 1996 |

151 |
Complex Analytic Coordinates in Almost Complex Manifolds,
- Newlander, Nirenberg
- 1957
(Show Context)
Citation Context ...|g∞ = O(e λt). We are now able to construct the desired extension of J by appealing to (an obvious localised version of) [12, Theorem 3.1], whose proof reduces the problem to the classical results of =-=[22, 23]-=-. Up until (A.2), this argument is a slight modification of Li’s approach; Li instead rewrites (A.1) as a comparison of J and J0 with respect to the smooth metric g̃0 = dρ 2 + ρ2dθ2 + gD on ∆ × B, whe... |

132 |
Métriques Kählériennes et fibrés holomorphes,” Ann
- Calabi
- 1979
(Show Context)
Citation Context ...ano manifold. For k ∈ N dividing c1(D), let M n be an AC Calabi-Yau manifold with asymptotic cone C = ( 1 k KD) × given by the Calabi ansatz. (i) If D = P2 and k = 1, then M = KD with Calabi’s metric =-=[3]-=-. (ii) If D = P1×P1 and k = 1, then either M = KD with one of Goto’s deformations of Calabi’s metric [8], or M = P3 \ quadric = T ∗RP3 with a Z2-quotient of Stenzel’s metric [30]. (iii) If D = P1×P1 a... |

131 | Uber Modifikationen und exzeptionnelle analytische Mengen, - GRAUERT - 1962 |

102 | Théorie de Hodge et géométrie algébrique complexe, Cours Spécialisés 10, Société Mathématique de France, - Voisin - 2002 |

101 |
Elliptic differential operators on noncompact manifolds
- Lockhart, McOwen
- 1985
(Show Context)
Citation Context ... Hölder spaces is that there exists a satisfactory elliptic theory associated with the mapping (2.5), which we now summarize. Our reference is Marshall [45]; a major earlier paper by Lockhart-McOwen =-=[43]-=- develops the theory in weighted Sobolev spaces. Theorem 2.9 ([45, Theorem 6.10]). Define the set of exceptional weights P := { −m− 2 2 ± √ (m− 2)2 4 + µ : µ ≥ 0 is an eigenvalue of ∆L } . (2.6) Then ... |

101 |
Complete Kähler manifolds with zero Ricci curvature
- Tian, Yau
- 1991
(Show Context)
Citation Context ...ler-Einstein manifolds, then at least heuristically, these are precisely the ones that correspond to noncollapsed isolated singularities. Besides [17], the foundational paper in this area is Tian-Yau =-=[53]-=-, but see also [27, 36, 56]. Remark 1.1. Ricci-flat Kähler manifolds of Euclidean volume growth whose tangent cones are not smooth, and in fact not even products of smooth cones, exist as well: Joyce... |

95 | Ricci-flat metrics, harmonic forms and brane resolutions
- Cvetic, Gibbons, et al.
- 2003
(Show Context)
Citation Context ...The proof of Proposition 5.11 is an explicit computation based on the following lemma. One could also use the presentation of the Stenzel metric in terms of left-invariant 1-forms on SO(n+1) given in =-=[21]-=-; compare in particular the shapes of (5.8) and [21, (2.34)]. Lemma 5.14 (Stenzel [52]). Up to scaling, the unique SO(n + 1)-invariant AC Calabi-Yau metric g on the standard smoothing of the ordinary ... |

91 | Sasakian Geometry,
- Boyer, Galicki
- 2008
(Show Context)
Citation Context ...t least when α is purely polynomially behaved and does not contain any log terms) the rate of α is really the infimum of all λ for which this holds. 1.3.2. Kähler and Calabi-Yau cones. Boyer-Galicki =-=[10]-=- is a comprehensive reference here. Definition 1.7. A Kähler cone is a Riemannian cone (C, g0) such that g0 is Kähler, together with a choice of g0-parallel complex structure J0. This will in fact o... |

87 | Numerical characterization of the Kahler cone of a compact Kahler manifold.
- Demailly, Paun
- 2004
(Show Context)
Citation Context ... know, these classes may very well be characterized by the inequalities ∫ Y ω k > 0 for all k-dimensional (k > 0) compact analytic subsets Y ⊂M , as is of course the case on compact Kähler manifolds =-=[22]-=-. Our construction of AC Calabi-Yau metrics relies on a slightly more restrictive notion of Kähler class than this; compare Definition 2.3. However, in all the examples that we discuss in this paper,... |

85 |
Versal deformations and algebraic
- Artin
- 1974
(Show Context)
Citation Context ...to Kronheimer’s. Let (M,g,Ω) be an AC Calabi-Yau surface, so that the asymptotic cone (C, g0,Ω0) is isomorphic to C2/Γ with its standard metric and holomorphic volume form for some finite group Γ ⊂ SU=-=(2)-=- acting freely on S3. Then the family (M, tg, tΩ) (t ∈ R+) converges to (C, g0,Ω0) in the Gromov-Hausdorff sense as t→ 0. We would now like to upgrade this Gromov-Hausdorff family to an algebraic one,... |

83 | Holomorphic Morse Inequalities and Bergman Kernels, - Ma, Marinescu - 2007 |

80 |
The Lefschetz theorem on hyperplane sections,
- Andreotti, Frankel
- 1959
(Show Context)
Citation Context .... Example 4.10. The small resolution of the conifold from Example 4.4 is recovered by taking G to be SU(2)× SU(2). This group, being of rank 2, has two flag manifolds: P1 × P1 and P1. Example 4.11. SU=-=(3)-=- also has two flag manifolds: the maximal one, SU(3)/T 2 = P(T ∗P2) (of Fano index 2, as for all groups G), and P2. The resulting 1-parameter family of AC Calabi-Yau metrics on T ∗P2 is originally due... |

73 |
Verschwindungssätze für analytische Kohomologiegruppen auf Komplexen Räume.
- Grauert, Riemenschneider
- 1970
(Show Context)
Citation Context ...6], so that the vanishing follows from [11, Satz 2.1]. (Both versions of this argument require X to be smooth as we need the sheaf of sections of [D] to be locally free in order to have that · = ⊗ in =-=[11, 25]-=-.) That the singularities of Y \ p(D) are canonical is clear by definition. (iii) By [21, Theorem 2.2.18], the Hodge theorem holds on X. Thus, H2(X) = H1,1(X) by (ii). As in [6, Proposition 2.5], it f... |

71 |
Some integration problems in almost-complex and complex
- Nijenhuis, Woolf
- 1963
(Show Context)
Citation Context ...|g∞ = O(e λt). We are now able to construct the desired extension of J by appealing to (an obvious localised version of) [12, Theorem 3.1], whose proof reduces the problem to the classical results of =-=[22, 23]-=-. Up until (A.2), this argument is a slight modification of Li’s approach; Li instead rewrites (A.1) as a comparison of J and J0 with respect to the smooth metric g̃0 = dρ 2 + ρ2dθ2 + gD on ∆ × B, whe... |

60 | Fano manifolds, contact structures, and quaternionic geometry - LeBrun - 1995 |

57 |
Ricci-flat metrics on the complexification of a compact rank one symmetric space,” manuscripta mathematica 80
- Stenzel
- 1993
(Show Context)
Citation Context ...D with Calabi’s metric [3]. (ii) If D = P1×P1 and k = 1, then either M = KD with one of Goto’s deformations of Calabi’s metric [8], or M = P3 \ quadric = T ∗RP3 with a Z2-quotient of Stenzel’s metric =-=[30]-=-. (iii) If D = P1×P1 and k = 2, then C is the 3-fold ordinary double point, and either M = T ∗S3, the smoothing of C, with Stenzel’s metric [30], or else M = OP1(−1) ⊕2, the small resolution of C, wit... |

57 | Theory of Stein spaces - Grauert, Remmert - 1979 |

54 |
Kählerian coset spaces of semi-simple Lie groups
- Borel
- 1954
(Show Context)
Citation Context ...ubgroups P1 ( P2 such that P2/P1 = P k−1, where k divides c1(G/P1). The main point here is that all flag manifolds G/P are Fano with a G-invariant Kähler-Einstein metric [6, §8], and Hodd(G/P,Z) = 0 =-=[8]-=-, so that the bundle G/P1 → G/P2 can be written as P(E) 16 Ronan J. Conlon and Hans-Joachim Hein for some vector bundle E → G/P2 by [32, p. 515]. The condition on c1 is combinatorially checkable, alth... |

50 | Lokale topologische Eigenschaften komplexer Rfiume. - Hamm - 1971 |

48 |
Algebraic methods in the global theory of complex spaces
- Banica, Stanasila
- 1976
(Show Context)
Citation Context ...ample 4.11 above. Remark 4.14. Flag manifolds and their characteristic classes are a classical topic in topology. For example, Borel-Hirzebruch [9, p. 340] noticed that the Chern numbers c51 of the SU=-=(4)-=- flag manifolds P(T ∗P3) and P(TP3) are different. Thus, the complex structures of P(T ∗P3) and P(TP3) cannot be homotopic, even though their underlying smooth manifolds are trivially diffeomorphic. A... |

46 | Deformations of algebraic varieties with Gm action - Pinkham - 1974 |

45 | The versal deformation of an isolated toric Gorensteins singularity
- Altmann
(Show Context)
Citation Context ...i) to its normal cone. The affine cone in (i) is rigid, whereas the ones in (ii), (iii), (iv) have exactly one deformation, which is smooth. (For (i) this follows from [27], for (i), (ii), (iii) from =-=[1]-=-, and for (iii), (iv) from [13].) Alternatively, we can apply the classification theory of log-Fano varieties [28, Definition 2.1.1] to Y : in (i) and (ii), (Y,D) is a del Pezzo 3-fold of degree 9 and... |

45 |
On the cone structure at infinity of Ricci flat manifolds with Euclidean volume growth and quadratic curvature decay, Invent
- Cheeger, Tian
- 1994
(Show Context)
Citation Context ...er some length space. If Ric = 0, then this cone is likely to be the same for all sequences, so that M would be asymptotically conical in a Gromov-Hausdorff sense. This was proved by Cheeger and Tian =-=[17]-=-, assuming |Rm| = O(r−2) (so that the links of all bona fide tangent cones are smooth) and an integrability condition. In the course of their proof, they in fact establish C∞ convergence to one such c... |

44 | and R.Thomas, Weighted projective embeddings, stability of orbifolds and constant scalar curvature Kähler metrics - Ross |

38 | Primitive Calabi-Yau threefolds
- Gross
- 1997
(Show Context)
Citation Context ...er a 3-dimensional compact Calabi-Yau variety Y with isolated singularities. It is reasonably well understood under what conditions Y can or cannot be deformed to a smooth Calabi-Yau 3-fold; see e.g. =-=[25, 33]-=-. On the other hand, if all the singularities of Y are locally isomorphic to Calabi-Yau cones Ci, then Y is widely expected to admit Calabi-Yau metrics with conical singularities modelled on the Ci [1... |

32 |
Simultaneous resolution of threefold double
- Friedman
- 1986
(Show Context)
Citation Context ...er a 3-dimensional compact Calabi-Yau variety Y with isolated singularities. It is reasonably well understood under what conditions Y can or cannot be deformed to a smooth Calabi-Yau 3-fold; see e.g. =-=[25, 33]-=-. On the other hand, if all the singularities of Y are locally isomorphic to Calabi-Yau cones Ci, then Y is widely expected to admit Calabi-Yau metrics with conical singularities modelled on the Ci [1... |

29 | Comparison theorem for Kähler manifolds and positivity of spectrum - Li, Wang |

28 |
Degeneration of Riemannian metrics under Ricci curvature bounds, Lezioni Fermiane, Scuola Normale Superiore
- Cheeger
- 2001
(Show Context)
Citation Context ...imal rate for the Stenzel metric on T ∗Sn is −2 n n−1 . 1. Introduction 1.1. Background. Consider a complete Riemannian manifold (M,g) with Ric ≥ 0 and Euclidean volume growth. Cheeger-Colding theory =-=[16]-=- implies that each blowdown sequence (M,λig), λi → 0, subconverges in the pointed Gromov-Hausdorff sense to the cone over some length space. If Ric = 0, then this cone is likely to be the same for all... |

25 | Affine open subsets of algebraic varieties and ample divisors - Goodman - 1969 |

25 | Resolutions of non-regular Ricci-flat Kahler cones
- Martelli, Sparks
- 2009
(Show Context)
Citation Context ... Joyce’s foundational work on the ALE case [36]. Note that the classes represented by Calabi’s metrics are compactly supported, and H2(M) = H2c (M) anyway if M is any resolution of C n/Γ. We refer to =-=[36, 46, 56, 57]-=- for many new examples beyond these, including crepant resolutions of irregular Calabi-Yau cones. 14 Ronan J. Conlon and Hans-Joachim Hein 4.2. Goto’s theorem. What Section 4.1 leaves open is whether ... |

20 |
On the versal deformation of a complex space with an isolated singularity
- Kas, Schlessinger
- 1972
(Show Context)
Citation Context ... then C is the n-fold ordinary double point, and M = T ∗Sn, the smoothing of C, with Stenzel’s metric [30]. 1Standard results in deformation theory (see [2, Example 4.5] in the algebraic category and =-=[13]-=- or [10, p.198, Theorem] in the analytic category) all require shrinking the total space Y of the given deformation in order to be able to pull Y back from the versal deformation. We can undo this shr... |

18 | Invarianten quasihomogener vollstandiger Durchschnitten, - Hamm - 1978 |

18 | Deformations of special Lagrangian submanifolds, D.Phil
- Marshall
- 2002
(Show Context)
Citation Context ... (M). (2.5) The main point of defining weighted Hölder spaces is that there exists a satisfactory elliptic theory associated with the mapping (2.5), which we now summarize. Our reference is Marshall =-=[45]-=-; a major earlier paper by Lockhart-McOwen [43] develops the theory in weighted Sobolev spaces. Theorem 2.9 ([45, Theorem 6.10]). Define the set of exceptional weights P := { −m− 2 2 ± √ (m− 2)2 4 + µ... |

17 |
Hyper-Kähler metrics on cotangent bundles of Hermitian symmetric spaces
- Biquard, Gauduchon
- 1995
(Show Context)
Citation Context ...hose tangent cones are not smooth, and in fact not even products of smooth cones, exist as well: Joyce’s QALE spaces [36] are desingularizations of very general flat orbifolds Cn/Γ. Biquard-Gauduchon =-=[7]-=- wrote down explicit hyper-Kähler examples whose tangent cones are realized as nilpotent orbit closures in sl(N,C). The following three basic AC Ricci-flat examples are helpful to keep in mind throug... |

15 |
Sasaki–Einstein manifolds. Surveys in differential geometry. Volume XVI. Geometry of special holonomy and related topics, 265–324,
- Sparks
- 2011
(Show Context)
Citation Context ...Kähler manifolds with torsion canonical bundle to [20]. 1.3.3. Calabi ansatz. The construction of Calabi-Yau cones, or Sasaki-Einstein manifolds, is in itself a highly nontrivial problem. See Sparks =-=[51]-=- for an excellent recent survey. The most elementary construction, originating in Calabi’s paper [12], states that regular CalabiYau cones are classified by Kähler-Einstein Fano manifolds; see LeBrun... |

14 |
Coevering, “A Construction of Complete Ricci-flat Kähler Manifolds,” arXiv:0803.0112 [math.DG
- van
(Show Context)
Citation Context ...se for less than maximal volume growth. The same idea turns out to be useful in other settings as well, e.g. when dealing with isolated conical singularities. Examples: General picture. Van Coevering =-=[54]-=- pointed out that all AC Kähler manifolds can be made Stein by contracting compact analytic sets. We then have the following rough general picture, which we will flesh out in Sections 4–5 by looking ... |

9 | Asymptotically cylindrical Calabi–Yau manifolds. arXiv math/1212.6929
- Haskins, Hein, et al.
(Show Context)
Citation Context ... unlike the construction of X as a complex orbifold in [20] (compare Appendix A), our proof that X is Kähler is quite different in spirit from the treatment of the asymptotically cylindrical case in =-=[12]-=-. 4 Ronan J. Conlon and Hans-Joachim Hein Proof of Theorem 2.1 in the smooth case. (i) This was already shown in the proof of [6, Proposition 2.4], based on Grauert’s generalisation of the Kodaira emb... |

9 | On optimal 4-dimensional metrics
- LeBrun, Maskit
(Show Context)
Citation Context ...ith respect to g̃0. Finally, we point out that both arguments obviously extend to the orbifold setting. Alternatively, in order to prove Corollary C, we could also use a compactification theorem from =-=[19]-=-, which relies on twistor theory. In fact, [19, Lemma 4.1] asserts that if the cone (L \ 0, ω0) is flat of dimension 2 (hence is of the form C2/Γ for some finite group Γ ⊂ U(2) acting freely on S3 ⊂ C... |

9 | On rational singularities in dimensions - Burns - 1974 |

9 | Desingularizations of Calabi-Yau 3-folds with conical singularities
- Chan
(Show Context)
Citation Context ...omorphic to Calabi-Yau cones Ci, then Y is widely expected to admit Calabi-Yau metrics with conical singularities modelled on the Ci [15, Definition 4.6]. Assuming the existence of such metrics, Chan =-=[15]-=- used gluing arguments to prove that Y is indeed smoothable if there exist AC Calabi-Yau manifolds Mi asymptotic to Ci at rate strictly faster than −3. (2a) Our computations yield the critical rate −3... |

7 |
Calabi-Yau structures and Einstein-Sasakian structures on crepant resolutions of isolated singularities,” arXiv:0906.5191v2 [math.DG
- Goto
(Show Context)
Citation Context ...1 k KD) × given by the Calabi ansatz. (i) If D = P2 and k = 1, then M = KD with Calabi’s metric [3]. (ii) If D = P1×P1 and k = 1, then either M = KD with one of Goto’s deformations of Calabi’s metric =-=[8]-=-, or M = P3 \ quadric = T ∗RP3 with a Z2-quotient of Stenzel’s metric [30]. (iii) If D = P1×P1 and k = 2, then C is the 3-fold ordinary double point, and either M = T ∗S3, the smoothing of C, with Ste... |

7 | Symmetries, quotients and Kähler-Einstein metrics
- Arezzo, Ghigi, et al.
(Show Context)
Citation Context ...e distance from the apex in the metric completion. Suppose that we are given a Riemannian cone (C, g0) as above. Let (r, x) be polar coordinates on C, where x ∈ L, and for t > 0, define a map νt : L× =-=[1, 2]-=- ∋ (r, x) 7→ (tr, x) ∈ L× [t, 2t]. One checks that ν∗t (g0) = t2g0 and ν∗t ◦ ∇0 = ∇0 ◦ ν∗t , where ∇0 is the Levi-Civita connection of g0. Using these facts, one can prove the following basic lemma wh... |

6 | Harmonic functions of linear growth on Kähler manifolds with nonnegative Ricci curvature
- Li
- 1995
(Show Context)
Citation Context ...nifold with Ric ≥ 0 is of necessity pluriharmonic (Corollary 3.9). The proof makes heavy use of the AC structure (the key ingredient here is a lemma from Cheeger-Tian [17]); contrary to a claim in Li =-=[41]-=-, the statement is in fact false for less than maximal volume growth. The same idea turns out to be useful in other settings as well, e.g. when dealing with isolated conical singularities. Examples: G... |

6 | Rigidity for multi-Taub-NUT metrics,
- Minerbe
- 2011
(Show Context)
Citation Context ...This is false: Consider the Taub-NUT manifold, which is complete hyper-Kähler of real dimension 4 with cubic volume growth, with a triholomorphic Killing field X that rotates the circles at infinity =-=[47]-=-. Fix a parallel complex structure J and define u by du = X xω. Then u is harmonic of linear growth (asymptotic to the linear function on R3 determined by J ∈ S2 ⊂ R3), but not J-pluriharmonic because... |

5 |
Characterizing Moisezon spaces by almost positive coherent analytic sheaves,
- RIEMENSCHNEIDER
- 1971
(Show Context)
Citation Context ...6], so that the vanishing follows from [11, Satz 2.1]. (Both versions of this argument require X to be smooth as we need the sheaf of sections of [D] to be locally free in order to have that · = ⊗ in =-=[11, 25]-=-.) That the singularities of Y \ p(D) are canonical is clear by definition. (iii) By [21, Theorem 2.2.18], the Hodge theorem holds on X. Thus, H2(X) = H1,1(X) by (ii). As in [6, Proposition 2.5], it f... |

5 |
Harmonic functions on manifolds of nonnegative Ricci curvature
- Donnelly
(Show Context)
Citation Context ...ū0 onM . Then ∆gū0 ∈ C∞λ−2−ε0(M). Since we have λ−2−ε0 > −m, Theorem 2.11 tells us that there exists v of rate at most λ− ε0 such that ∆gv = −∆gū0. (This type of argument is quite common; see e.g. =-=[18, 24]-=- for very similar results and applications.) The following is then the promised extension of Lemma 3.4 from cones to AC manifolds. Notice that the Bochner formula already tells us that M does not ad... |

5 | Desingularizing isolated conical singularities: uniform estimates via weighted Sobolev spaces - Pacini |

5 | Exceptional holonomy and Einstein metrics constructed from Aloff-Wallach spaces
- Reidegeld
- 2011
(Show Context)
Citation Context ...-Yau metrics on T ∗P2 is originally due to Calabi [12]; these metrics are in fact hyper-Kähler. Conjecture 4.8 was motivated by the fact that such contractions were recently constructed in this case =-=[5, 49]-=-. The example of SU(3) can be generalized to higher dimensions in two different ways. We refer to [6, 8.111] for computations involving SU(n+1) flags. The basic tool here is to label the space of all ... |

4 |
E.G.: The Spin(7)-structures on complex line bundles and explicit Riemannian metrics with SU(4)-holonomy
- Bazaikin, Malkovich
(Show Context)
Citation Context ...-Yau metrics on T ∗P2 is originally due to Calabi [12]; these metrics are in fact hyper-Kähler. Conjecture 4.8 was motivated by the fact that such contractions were recently constructed in this case =-=[5, 49]-=-. The example of SU(3) can be generalized to higher dimensions in two different ways. We refer to [6, 8.111] for computations involving SU(n+1) flags. The basic tool here is to label the space of all ... |

4 | Einstein manifolds, Ergebnisse d - Besse |

4 | de la Ossa, Comments on conifolds, Nuclear Phys - Candelas, X - 1990 |

4 |
Harmonic functions of polynomial growth on certain complete manifolds
- Christiansen, Zworski
- 1996
(Show Context)
Citation Context ...ū0 onM . Then ∆gū0 ∈ C∞λ−2−ε0(M). Since we have λ−2−ε0 > −m, Theorem 2.11 tells us that there exists v of rate at most λ− ε0 such that ∆gv = −∆gū0. (This type of argument is quite common; see e.g. =-=[18, 24]-=- for very similar results and applications.) The following is then the promised extension of Lemma 3.4 from cones to AC manifolds. Notice that the Bochner formula already tells us that M does not ad... |

2 | flops, minimal models, etc”. In: Surveys in differential geometry - “Flips - 1990 |

2 |
On the quasi-asymptotically locally Euclidean geometry of Nakajima’s metric
- Carron
(Show Context)
Citation Context ...ε) cannot be relaxed to η = O(1). Theorem 3.1 is also false for volume growth 2n − 1 in place of 2n; the Taub-NUT metric will serve as a useful counterexample throughout the proof. Remark 3.3. Carron =-=[14]-=- proved a Calabi-Yau type uniqueness theorem, assuming an intermediate rate of decay, on the Hilbert scheme of N points in C2 with a QALE hyper-Kähler metric. It would be interesting to see whether t... |

2 | Weighted Sobolev inequalities under lower Ricci curvature bounds - Hein |

2 |
On the New Explicit Riemannian Metrics with Holonomy Group SU(4
- Malkovich
(Show Context)
Citation Context ...s to choose an element of ∧3,0C3. In any case, the result is that we recover Calabi’s hyper-Kähler metrics on T ∗Pn for every n ≥ 2. Conjecture 4.8 is in fact known to hold true in this case as well =-=[44]-=-, generalizing the work for n = 2 mentioned in Example 4.11 above. Remark 4.14. Flag manifolds and their characteristic classes are a classical topic in topology. For example, Borel-Hirzebruch [9, p. ... |

2 |
Kähler metrics on crepant resolutions of Kähler cones
- Ricci-flat
(Show Context)
Citation Context ...es k are in fact µ-almost compactly supported for some µ < 0. If k ∈ H2c (M), then µ = −∞ works and we obtain AC Calabi-Yau metrics with leading term i∂∂̄r2−2n. This is van Coevering’s main result in =-=[55, 56]-=- and contains Joyce’s foundational work on the ALE case [36]. Note that the classes represented by Calabi’s metrics are compactly supported, and H2(M) = H2c (M) anyway if M is any resolution of C n/Γ.... |

1 | die Deformation isolierter Singularitäten analytischer - Über - 1972 |

1 |
Remarks on deformation of cones and neighborhoods of ample divisors
- Li
(Show Context)
Citation Context ...nt resolutions of K×D whose exceptional set is not pure-dimensional). Acknowledgments. We are grateful to Mark Haskins for many helpful discussions over the years, to Chi Li for sending us a draft of =-=[20]-=-, to Richard Thomas for explaining [2, Example 4.5] to us, and to Jeff Viaclovsky for pointing out the compactification theorem [19, Lemma 4.1]. 2. Proof of Theorem A It is clear that Ω extends to a m... |