PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo
2011 | 9 | 3 | 535-557
Tytuł artykułu

Hyperholomorphic connections on coherent sheaves and stability

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let M be a hyperkähler manifold, and F a reflexive sheaf on M. Assume that F (away from its singularities) admits a connection ▿ with a curvature Θ which is invariant under the standard SU(2)-action on 2-forms. If Θ is square-integrable, such sheaf is called hyperholomorphic. Hyperholomorphic sheaves were studied at great length in [21]. Such sheaves are stable and their singular sets are hyperkähler subvarieties in M. In the present paper, we study sheaves admitting a connection with SU(2)-invariant curvature which is not necessary L 2-integrable. We show that such sheaves are polystable.
Bibliografia
  • [1] Bando S., Siu Y.-T, Stable sheaves and Einstein-Hermitian metrics, In: Geometry and Analysis on Complex Manifolds, Festschrift for Professor S. Kobayashi’s 60th Birthday, World Scientific, Singapore, 1994, 39–59
  • [2] Bartocci C., Bruzzo, U., Hernández Ruipérez D., A hyperkähler Fourier transform, Differential Geom. Appl., 1998, 8(3), 239–249 http://dx.doi.org/10.1016/S0926-2245(98)00009-6
  • [3] Beauville A., Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom., 1983, 18(4), 755–782
  • [4] Besse A.L., Einstein Manifolds, Ergeb. Math. Grenzgeb., 10, Springer, New York, 1987
  • [5] Calabi E., Métriques kählériennes et fibrès holomorphes, Ann. Sci. École Norm. Sup., 1979, 12(2), 269–294
  • [6] Demailly J.-P., L 2 vanishing theorems for positive line bundles and adjunction theory, In: Transcendental Methods of Algebraic P eometry, Cetraro, July 1994, Lecture Notes in Math., 1646, Springer, Berlin, 1996, 1–97 http://dx.doi.org/10.1007/BFb0094302
  • [7] Demailly J.-P., Complex Analytic and Algebraic Geometry, book available at http://www-fourier.ujf-grenoble.fr/~demailly/books.html
  • [8] El Mir H., Sur le prolongement des courants positifs fermés, Acta Math., 1984, 153(1–2), 1–45 http://dx.doi.org/10.1007/BF02392374
  • [9] Griffiths Ph., Harris J., Principles of Algebraic Geometry, Pure Appl. Math. (N.Y.), John Wiley & Sons, New York-Chichester-Brisbane-Toronto, 1978
  • [10] Kaledin D, Verbitsky M., Non-Hermitian Yang-Mills connections, Selecta Math. (N.S.), 1998, 4(2), 279–320 http://dx.doi.org/10.1007/s000290050033
  • [11] Kobayashi S., Differential Geometry of Complex Vector Bundles, Publ. Math. Soc. Japan, 15, Princeton University Press, Princeton, 1987
  • [12] Okonek Ch., Schneider M., Spindler H., Vector Bundles on Complex Projective Spaces, Progr. Math., 3, Birkhäuser, Boston, 1980
  • [13] Scheja G., Riemannsche Hebbarkeitssätze für Cohomologieklassen, Math. Ann., 1961, 144, 345–360 http://dx.doi.org/10.1007/BF01470506
  • [14] Sibony N., Quelques problèmes de prolongement de courants en analyse complexe, Duke Math. J., 1985, 52(1), 157–197 http://dx.doi.org/10.1215/S0012-7094-85-05210-X
  • [15] Simpson CT., Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc, 1988, 1(4), 867–918 http://dx.doi.org/10.1090/S0894-0347-1988-0944577-9
  • [16] Skoda H., Prolongement des courants positifs fermés de masse finie, Invent. Math., 1982, 66(3), 361–376 http://dx.doi.org/10.1007/BF01389217
  • [17] Uhlenbeck K., Yau ST., On the existence of Hermitian-Yang-Mills connections in stable vector bundles, In: Frontiers of the Mathematical Sciences, New York, 1985, Comm. Pure Appl. Math., 1986, 39(S1), S257–S293
  • [18] Verbitsky M., Tri-analytic subvarieties of hyper-Kaehler manifolds, Geom. Funct. Anal., 1995, 5(1), 92–104 http://dx.doi.org/10.1007/BF01928217
  • [19] Verbitsky M., Hyper-Kähler embeddings and holomorphic symplectic geometry I, J. Algebraic Geom., 1996, 5(3), 401–413
  • [20] Verbitsky M., Hyperholomorphic bundles over a hyper-Kähler manifold, J. Algebraic Geom., 1996, 5(4), 633–669
  • [21] Verbitsky M., Hyperholomorphic sheaves and new examples of hyperkähler manifolds, In: Kaledin D., Verbitsky M., Hyperkahler Manifolds, Math. Phys. (Somerville), 12, International Press, Somerville, 1999, first part of the book
  • [22] Verbitsky M., Hypercomplex varieties, Comm. Anal. Geom., 1999, 7(2), 355–396
  • [23] Verbitsky M., Plurisubharmonic functions in calibrated geometry and q-convexity, Math. Z., 2010, 264(4), 939–957 http://dx.doi.org/10.1007/s00209-009-0498-7
  • [24] Verbitsky M., Positive forms on hyperkähler manifolds, Osaka J. Math., 2010, 47(2), 353–384
  • [25] Yau ST., On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I., Comm. Pure Appl. Math., 1978, 31(3), 339–411 http://dx.doi.org/10.1002/cpa.3160310304
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-011-0016-0
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.