Skip to main content Accessibility help
×
Home

EXISTENCE AND COMPACTNESS THEORY FOR ALE SCALAR-FLAT KÄHLER SURFACES

  • JIYUAN HAN (a1) and JEFF A. VIACLOVSKY (a2)

Abstract

Our main result in this article is a compactness result which states that a noncollapsed sequence of asymptotically locally Euclidean (ALE) scalar-flat Kähler metrics on a minimal Kähler surface whose Kähler classes stay in a compact subset of the interior of the Kähler cone must have a convergent subsequence. As an application, we prove the existence of global moduli spaces of scalar-flat Kähler ALE metrics for several infinite families of Kähler ALE spaces.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      EXISTENCE AND COMPACTNESS THEORY FOR ALE SCALAR-FLAT KÄHLER SURFACES
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      EXISTENCE AND COMPACTNESS THEORY FOR ALE SCALAR-FLAT KÄHLER SURFACES
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      EXISTENCE AND COMPACTNESS THEORY FOR ALE SCALAR-FLAT KÄHLER SURFACES
      Available formats
      ×

Copyright

This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

References

Hide All
[AV12]Ache, A. G. and Viaclovsky, J. A., ‘Obstruction-flat asymptotically locally Euclidean metrics’, Geom. Funct. Anal. 22(4) (2012), 832877.
[Aku12]Akutagawa, K., ‘Computations of the orbifold Yamabe invariant’, Math. Z. 271(3–4) (2012), 611625.
[AB04]Akutagawa, K. and Botvinnik, B., ‘The Yamabe invariants of orbifolds and cylindrical manifolds, and L 2 -harmonic spinors’, J. Reine Angew. Math. 574 (2004), 121146.
[And89]Anderson, M. T., ‘Ricci curvature bounds and Einstein metrics on compact manifolds’, J. Amer. Math. Soc. 2(3) (1989), 455490.
[ALM14]Arezzo, C., Lena, R. and Mazzieri, L., ‘On the resolution of extremal and constant scalar curvature Kähler orbifolds’, Int. Math. Res. Not. IMRN 2016(21) (2016), 64156452.
[AP06]Arezzo, C. and Pacard, F., ‘Blowing up and desingularizing constant scalar curvature Kähler manifolds’, Acta Math. 196(2) (2006), 179228.
[Art74]Artin, M., ‘Algebraic construction of Brieskorn’s resolutions’, J. Algebra 29 (1974), 330348.
[AI08]Ashikaga, T. and Ishizaka, M., Another form of the reciprocity law of Dedekind sum, Hokkaido University EPrints Server, no. 908, http://eprints3.math.sci.hokudai.ac.jp/1849/, 2008.
[Ban90]Bando, S., ‘Bubbling out of Einstein manifolds’, Tohoku Math. J. (2) 42(2) (1990), 205216.
[BKN89]Bando, S., Kasue, A. and Nakajima, H., ‘On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth’, Invent. Math. 97(2) (1989), 313349.
[Bar86]Bartnik, R., ‘The mass of an asymptotically flat manifold’, Comm. Pure Appl. Math. 39(5) (1986), 661693.
[BC94]Behnke, K. and Christophersen, J. A., ‘M-resolutions and deformations of quotient singularities’, Amer. J. Math. 116(4) (1994), 881903.
[BR15]Biquard, O. and Rollin, Y., ‘Smoothing singular extremal Kähler surfaces and minimal Lagrangians’, Adv. Math. 285 (2015), 9801024.
[Bur86]Burns, D., Twistors and harmonic maps, Talk in Charlotte, N.C., October 1986.
[Cal79]Calabi, E., ‘Métriques kählériennes et fibrés holomorphes’, Ann. Sci. Éc. Norm. Supér. (4) 12(2) (1979), 269294.
[CS04]Calderbank, D. M. J. and Singer, M. A., ‘Einstein metrics and complex singularities’, Invent. Math. 156(2) (2004), 405443.
[CLW08]Chen, X., Lebrun, C. and Weber, B., ‘On conformally Kähler, Einstein manifolds’, J. Amer. Math. Soc. 21(4) (2008), 11371168.
[Elk74]Elkik, R., ‘Solution d’équations au-dessus d’anneaux henséliens’, inQuelques problèmes de modules (Sém. Géom. Anal., École Norm. Supér., Paris, 1971–1972), Astérisque, 16 (Soc. Math. France, Paris, 1974), 116132.
[Gra72]Grauert, H., ‘über die Deformation isolierter Singularitäten analytischer Mengen’, Invent. Math. 15 (1972), 171198.
[GR84]Grauert, H. and Remmert, R., Coherent Analytic Sheaves, (Springer, Berlin, 1984).
[GK82]Greene, R. E. and Krantz, S. G., ‘Deformation of complex structures, estimates for the ̄ equation, and stability of the Bergman kernel’, Adv. Math. 43(1) (1982), 186.
[GLS07]Greuel, G.-M., Lossen, C. and Shustin, E., Introduction to singularities and deformations, Springer Monographs in Mathematics (Springer, Berlin, 2007).
[HV16]Han, J. and Viaclovsky, J. A., ‘Deformation theory of scalar-flat Kähler ALE surfaces’, Amer. J. Math. 141(6) (2019), 15471589.
[HL75]Harvey, F. R. and Blaine Lawson, H. Jr., ‘On boundaries of complex analytic varieties. I’, Ann. of Math. (2) 102(2) (1975), 223290.
[Heb96]Hebey, E., Sobolev Spaces on Riemannian Manifolds, Lecture Notes in Mathematics, 1635 (Springer, Berlin, 1996).
[HL16]Hein, H.-J. and LeBrun, C., ‘Mass in Kähler geometry’, Comm. Math. Phys. 347(1) (2016), 183221.
[HRŞ16]Hein, H.-J., Rasdeaconu, R. and Şuvaina, I., ‘On the classification of ALE Kähler manifolds’, Int. Math. Res. Not. IMRN, to appear. Preprint, 2016, arXiv:1610.05239.
[Hir53]Hirzebruch, F., ‘Über vierdimensionale Riemannsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen’, Math. Ann. 126 (1953), 122.
[Hit97]Hitchin, N. J., ‘Einstein metrics and the eta-invariant’, Boll. Unione. Mat. Ital. B (7) 11(2) (1997), 95105 (suppl.).
[Joy00]Joyce, D. D., Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs (Oxford University Press, Oxford, 2000).
[Kaw81]Kawasaki, T., ‘The index of elliptic operators over V-manifolds’, Nagoya Math. J. 84 (1981), 135157.
[KSB88]Kollár, J. and Shepherd-Barron, N. I., ‘Threefolds and deformations of surface singularities’, Invent. Math. 91(2) (1988), 299338.
[Kro89]Kronheimer, P. B., ‘The construction of ALE spaces as hyper-Kähler quotients’, J. Differential Geom. 29(3) (1989), 665683.
[Lau79]Laufer, H. B., ‘Ambient deformations for exceptional sets in two-manifolds’, Invent. Math. 55(1) (1979), 136.
[LM08]LeBrun, C. and Maskit, B., ‘On optimal 4-dimensional metrics’, J. Geom. Anal. 18(2) (2008), 537564.
[LP07]Lee, Y. and Park, J., ‘A simply connected surface of general type with p g = 0 and K 2 = 2’, Invent. Math. 170(3) (2007), 483505.
[Lem92]Lempert, L., ‘On three-dimensional Cauchy–Riemann manifolds’, J. Amer. Math. Soc. 5(4) (1992), 923969.
[Lem94]Lempert, L., ‘Embeddings of three-dimensional Cauchy–Riemann manifolds’, Math. Ann. 300(1) (1994), 115.
[Li14]Li, C., ‘On sharp rates and analytic compactifications of asymptotically conical Kähler metrics’, Duke Math. J., to appear. Preprint, 2014, arXiv:1405.2433.
[LV15]Lock, M. T. and Viaclovsky, J. A., ‘Anti-self-dual orbifolds with cyclic quotient singularities’, J. Eur. Math. Soc. (JEMS) 17(11) (2015), 28052841.
[LV19]Lock, M. T. and Viaclovsky, J. A., ‘A smörgåsbord of scalar-flat Kähler ALE surfaces’, J. Reine Angew. Math. 746 (2019), 171208.
[Nak90]Nakajima, H., ‘Self-duality of ALE Ricci-flat 4-manifolds and positive mass theorem’, inRecent Topics in Differential and Analytic Geometry (Academic Press, Boston, MA, 1990), 385396.
[Nak94]Nakajima, H., ‘A convergence theorem for Einstein metrics and the ALE spaces’, inSelected Papers on Number Theory, Algebraic Geometry, and Differential Geometry, Amer. Math. Soc. Transl. Ser. 2, 160 (American Mathematical Society, Providence, RI, 1994), 7994.
[Nar62a]Narasimhan, R., ‘The Levi problem for complex spaces. II’, Math. Ann. 146 (1962), 195216.
[Nar62b]Narasimhan, R., ‘A note on Stein spaces and their normalisations’, Ann. Sc. Norm. Supér. Pisa (3) 16 (1962), 327333.
[Pet94]Peternell, T., ‘Pseudoconvexity, the Levi problem and vanishing theorems’, inSeveral Complex Variables, VII, Encyclopaedia Math. Sci., 74 (Springer, Berlin, 1994), 221257.
[Pin78]Pinkham, H., ‘Deformations of normal surface singularities with C action’, Math. Ann. 232(1) (1978), 6584.
[Rie74]Riemenschneider, O., ‘Deformationen von Quotientensingularitäten (nach zyklischen Gruppen)’, Math. Ann. 209 (1974), 211248.
[Ros63]Rossi, H., ‘Vector fields on analytic spaces’, Ann. of Math. (2) 78 (1963), 455467.
[Str10]Streets, J., ‘Asymptotic curvature decay and removal of singularities of Bach-flat metrics’, Trans. Amer. Math. Soc. 362(3) (2010), 13011324.
[Şuv12]Şuvaina, I., ‘ALE Ricci-flat Kähler metrics and deformations of quotient surface singularities’, Ann. Global Anal. Geom. 41(1) (2012), 109123.
[Tia90]Tian, G., ‘On Calabi’s conjecture for complex surfaces with positive first Chern class’, Invent. Math. 101(1) (1990), 101172.
[TV05a]Tian, G. and Viaclovsky, J., ‘Bach-flat asymptotically locally Euclidean metrics’, Invent. Math. 160(2) (2005), 357415.
[TV05b]Tian, G. and Viaclovsky, J., ‘Moduli spaces of critical Riemannian metrics in dimension four’, Adv. Math. 196(2) (2005), 346372.
[TV08]Tian, G. and Viaclovsky, J., ‘Volume growth, curvature decay, and critical metrics’, Comment. Math. Helv. 83(4) (2008), 889911.
[Via10]Viaclovsky, J., ‘Monopole metrics and the orbifold Yamabe problem’, Ann. Inst. Fourier (Grenoble) 60(7) (2010), 25032543.
[Wah79]Wahl, J. M., ‘Simultaneous resolution of rational singularities’, Compos. Math. 38(1) (1979), 4354.
[Wri12]Wright, E. P., ‘Quotients of gravitational instantons’, Ann. Global Anal. Geom. 41(1) (2012), 91108.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

MSC classification

EXISTENCE AND COMPACTNESS THEORY FOR ALE SCALAR-FLAT KÄHLER SURFACES

  • JIYUAN HAN (a1) and JEFF A. VIACLOVSKY (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed