Skip to main content
SpringerLink
Log in

A Problem List on Vector Bundles

  • Chapter
Book cover Complex Analysis and Geometry

Part of the book series: The University Series in Mathematics ((USMA))

  • 1042 Accesses

Abstract

I think that lists of open problems and questions are very useful, and I would like to see them in print more often (e.g., at the end of every Proceedings volume). Such lists stimulate research, and give more opportunities for researchers to disseminate their results among interested people. Furthermore, such lists can include relevant references, not all of which might be known to a young, isolated researcher.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. R. Adams, J. Hamad, and E. Previato, Isospectral Hamiltonian flows in finite and infinite dimension. I: Generalized Moser systems and moment maps into loop algebras, Comm. Math. Phys. 117, 451–500 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  2. V. Ancona and G. Ottaviani, Canonical resolutions of sheaves on Schubert and Brieskorn varieties, preprint (1989).

    Google Scholar 

  3. V. Ancona and G. Ottaviani, On the stability of special instanton bundles on p2n+1, preprint (1990).

    Google Scholar 

  4. I. Yu. Artiushkin, Four-dimensional Fano manifolds representable as P’-bundles, Vest. Mosk. Un-ta, Ser. 7, No.1, 12–16 (1987) (in Russian).

    Google Scholar 

  5. E. Ballico, On rank 2 vector bundles over an algebraic surface, Forum Math., to appear.

    Google Scholar 

  6. E. Ballico and R. Brussee, On the unbalance of vector bundles on a blown-up surface, preprint.

    Google Scholar 

  7. E. Ballico and L. Chiantini, Some properties of vector bundles on algebraic surfaces. Forum Math., to appear.

    Google Scholar 

  8. E. Ballico and L. Chiantini, Rank 2 bundles on surfaces and seminatural cohomology, preprint (1991).

    Google Scholar 

  9. C. Banica, Topologisch triviale holomorphe Vector bundel auf Pn(C), J. Reine Angew. Math. 344, 102–119(1983).

    MATH  MathSciNet  Google Scholar 

  10. C. Banica and J. Le Potier, Sur l’existence des fibres vectoriels holomorphes sur les surfaces, J. Reine Angew. Math. 378, 1–31 (1987).

    MATH  MathSciNet  Google Scholar 

  11. C. Banica and M. Putinar, On complex vector bundles on projective threefolds, Invent. Math. 88, 427–438 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  12. L. Barbieri Viale, K-coomologia e cicli algebrici locali, Comp. Rend. Acad. Sci. Paris 311, 223–228 (1990).

    MATH  MathSciNet  Google Scholar 

  13. W. Barth, Some properties of stable rank-2 vector bundles on P n , Math. Ann. 226, 125–150 (1977).

    Article  MATH  MathSciNet  Google Scholar 

  14. A. Beauville, Fibrés de rang 2 sur une courbe, fibré déterminant et fonctions theta, Bull. Soc. Math. France 116, 431–448 (1988).

    MATH  MathSciNet  Google Scholar 

  15. A. Beauville, Jacobiennes des courbes spectrales et systemes hamiltoniens completement integrables, Acta Math. 16, 211–235 (1990).

    Article  MathSciNet  Google Scholar 

  16. A. Beilinson, Coherent sheaves on P” and problems of linear algebra. English transl. in Funct. Anal. Appl. 12, 214–216 (1978).

    Article  MathSciNet  Google Scholar 

  17. G. Bohnhorst and H. Spindler, The stability of certain vector bundles on Pn, preprint, Osnabrukker (1990).

    Google Scholar 

  18. E. Brieskorn, Uber holomorphe P n -bundel uber P1, Math. Ann. 157, 343–357 (1965).

    Article  MATH  MathSciNet  Google Scholar 

  19. V. Brinzanescu and P. Flonder, Holomorphic 2-vector bundles on nonalgebraic 2-tori, J. Reine Angew. Math. 363, 47–58 (1985).

    MATH  MathSciNet  Google Scholar 

  20. J. L. Burchnall and T. W. Chaundy, Commutative ordinary differential operators, Proc. London Math. Soc. (2)21, 420–440 (1922);

    MathSciNet  Google Scholar 

  21. J. L. Burchnall and T. W. Chaundy, Commutative ordinary differential operators, Proc. R. Soc. London Ser. A 118, 557–583 (1928);

    Article  MATH  Google Scholar 

  22. J. L. Burchnall and T. W. Chaundy, Commutative ordinary differential operators, Proc. London Math. Soc. (2)134,471–485 (1932).

    Google Scholar 

  23. P. Dehornoy, Operateurs differentials et courbes elliptiques, Compositio Math. 43, 71–99 (1981).

    MATH  MathSciNet  Google Scholar 

  24. P. A. Deift, L. C. Li, T. Nanda, and C. Tomei, The Toda flow on a generic orbit is integrable, Comm. Pure Appl. Math. 39, 183–232 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  25. I. V. Demin, Three dimensional Fano manifolds representable as line fiberings, Izv. Akad. Nauk SSSR 44, No. 4 (1980); English transi, in Math. USSR Izv. 17; Addendum, Izv. Akad. Nauk SSSR 46, No. 3; English transi, in Math. USSR Izv. 20.

    MathSciNet  Google Scholar 

  26. U. V. Desale and S. Ramanan, Classification of vector bundles of rank 2 on hyperelliptic curves, Invent. Math. 38, 161–185 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  27. S. K. Donaldson, Polynomial invariants for smooth four-manifolds, Topology 29, 257–316 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  28. I. Ya. Dorfman and F. W. Nijhoff, On a 2 + 1-dimensional version of the Krichever-Novikov equation, INS Preprint No. 167 (1990).

    Google Scholar 

  29. J.-M. Drezet, Fibrés exceptionnels et suite spectrale de Beilinson généralizée sur P2, Math. Ann. 275, 25–48 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  30. J.-M. Drezet and J. Le Potier, Fibrés stables et fibrés exceptionnels sur P2, Ann, Sci. Ecole Norm. Supp. (4)18, 193–243 (1985).

    MATH  Google Scholar 

  31. G. Elencwajg and O. Forster, Bounding cohomology groups of vector bundles on Pn, Math. Ann. 246, 251–270 (1980).

    Article  MATH  MathSciNet  Google Scholar 

  32. G. Elencwajg and O. Forster, Vector bundles on manifolds without divisors and a theorem on deformations, Ann. Inst. Fourier 32, 25–51 (1983).

    Article  MathSciNet  Google Scholar 

  33. N. Ercolani and H. Flaschka, The geometry of the Hill equation and of the Neumann system, Philos. Trans. R. Soc. London Ser. A 315, 405–422 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  34. H. Flaschka, Towards an algebro-geometric interpretation of the Neumann system, Tohoku Math. J. 36, 407–426 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  35. H. Flenner, Restrictions of semistable bundles on projective varieties, Comment. Math. Helv. 59, 635–650 (1980).

    Article  MathSciNet  Google Scholar 

  36. W. Fulton, Ample vector bundles, Chern classes and numerical criteria, Invent. Math. 32, 171–178 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  37. B. van Geemen, Schottky-Jung relations and vector bundles on hyperelliptic curves, Math. Ann. 281, 431–449 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  38. A. L. Gorondentsev, Exceptional bundles on surfaces with a moving anticanonical class, Izv. Akad. Nauk SSSR Ser. Mat. 52, 740–757 (1988) (Russian); English transi, in Math. USSR Izv. 33 (1989).

    Google Scholar 

  39. A. L. Gorondentsev, Exceptional bundles on surfaces with a moving anticanonical class, Izv. Akad. Nauk SSSR Ser. Mat. 52, 740–757 (1988) (Russian); English transi, in Math. USSR Izv. 33 (1989).

    Google Scholar 

  40. A. L. Gorondentsev and A. N. Rudakov, Exceptional vector bundles on projective spaces, Duke Math. J. 54, 115–130 (1987).

    Article  MathSciNet  Google Scholar 

  41. F. A. Grunbaum, The Kadomtsev-Petviashvili equation : An elementary approach to the “rank-2” solutions of Krichever and Novikov, Phys. Lett. A 139, 146–150 (1989).

    Article  MathSciNet  Google Scholar 

  42. R. Hartshorne, Stable vector bundles of rank 2 on P3, Math. Ann. 238, 229–286 (1978).

    Article  MATH  MathSciNet  Google Scholar 

  43. R. Hartshorne, Algebraic vector bundles on projective spaces: A problem list, Topology 18, 117–128(1979).

    Article  MATH  MathSciNet  Google Scholar 

  44. R. Hartshorne, Stable reflexive sheaves. Ill, Math. Ann. 279, 517–554 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  45. R. Hartshorne and A. Hirschowitz, Nouvelles courbes de bon genre, Math. Ann. 280, 353–367 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  46. A. Hirschowitz, Existence de faisceaux réflexifs de rang deux sur P3 à bonne cohomologie, Inst. Hautes Etudes Sci. Publ. Math. 56, 105–137 (1988).

    MathSciNet  Google Scholar 

  47. A. Hirschowitz and K. Hulek, Complete families of stable vector bundles over P2, in Complex Analysis and Algebraic Geometry (Proc. Gottingen 1985), pp. 19–33, Lecture Notes in Math., Vol. 1194, Springer-Verlag, Berlin and New York (1986).

    Chapter  Google Scholar 

  48. N. Hitchin, Stable bundles and integrable systems, Duke Math. J. 54, 91–114 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  49. G. Horrocks, Vector bundles on the punctured spectrum of a local ring, Proc. London Math. Soc. (3) 14, 689–713 (1964).

    Article  MATH  MathSciNet  Google Scholar 

  50. K. Hulek and J. Le Potier, Sur l’espace de modules des faisceaux semistable de rang 2, de classes the Chern (0, 3) sur P2, Ann. Inst. Fourier 39, 251–292 (1989).

    Article  MATH  Google Scholar 

  51. K. Hulek and S. A. Stromme, Appendix to the paper “Complete families of stable vector bundles over P2,” in Complex Analysis and Algebraic Geometry (Proc. Gottingen 1985), pp. 34–40, Lecture Notes in Math., Vol. 1194, Springer-Verlag, Berlin and New York (1986).

    Chapter  Google Scholar 

  52. M. Idà and M. Manaresi, Rank-two vector bundles on Pn uniform with respect to some rational curves, Arch. Math. 51, 266–273 (1978).

    Article  Google Scholar 

  53. J. F. Jardine and V. P. Snaith (eds.), The Lake Louise problem session, in Algebraic K-Theory: Connections with Geometry and Topology, pp. 516–550, Nato ASI Series C, Vol. 279, Kluwer Academic, Boston (1989).

    Google Scholar 

  54. M. Kapranov, On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math. 92, 479–508 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  55. H. Kim, Stable vector bundles on Enriques surfaces, thesis.

    Google Scholar 

  56. H. Knorrer, Geodesics on the ellipsoid, Invent. Math. 59, 119–143 (1980).

    Article  MathSciNet  Google Scholar 

  57. I. M. Krichever and S. P. Novikov, Holomorphic fiberings and nonlinear equations. Finite zone solutions of rank 2, Soviet Math. Dokl. 20, 650–654 (1979).

    MATH  Google Scholar 

  58. N. M. Kumar, Smooth degeneration of complete intersection curves in positive characteristic, Invent. Math. 104, 313–319 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  59. G. A. Latham, Solutions of the KP equations associated to rank three commuting differential operators over a singular elliptic curve, Phys. D 41, 55–66 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  60. G. A. Latham, The computational approach to commuting ordinary differential operators of orders six and nine, ANU preprint CMA-R28–90.

    Google Scholar 

  61. J. Milne, Canonical models of (mixed) Shimura varieties and automorphic vector bundles, in Automorphic Forms, Shimura Varieties, and L-Functions, pp. 283–414, Academic Press, New York (1990).

    Google Scholar 

  62. I. O. Mokhov, Commuting differential operators of rank 3, and nonlinear differential equations, Math. USSR Izv. 35, 629–655 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  63. S. Mukai, Symplectic structure of the moduli space of sheaves on an Abelian or K3 surface, Invent. Math. 77, 101–116 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  64. S. Mukai, On the moduli space of bundles on K3 surfaces, in Vector Bundles on Algebraic Varieties, pp. 341–413, Oxford Univ. Press (1987).

    Google Scholar 

  65. S. Mukai, Duality between D(x) and D(x) with its applications to Picard sheaves, Nagoya Math. J. 81, 153–175 (1981).

    MATH  MathSciNet  Google Scholar 

  66. S. Mukai, Semi-homogeneous vector bundles on an Abelian variety, J. Math. Kyoto Univ. 18, 239–272 (1978).

    MATH  MathSciNet  Google Scholar 

  67. D. Mumford, An algebro-geometric construction of commuting operators and of solutions to the Toda-lattice equation, Korteweg-de Vries equation and related non-linear equations, Int. Symp. Algebraic Geometry (Kyoto, 1977), pp. 115–153, Kinokuniya Bookstore, Tokyo (1978).

    Google Scholar 

  68. M. S. Narasimhan and G. Trautmann, Compactification of MP3(0, 2) and Poncelet pairs of conics, Pacific J. Math. 145, 225–365 (1990).

    Article  MathSciNet  Google Scholar 

  69. C. Okonek and H. Spindler, Mathematical instanton bundles on p2w+1, J. Reine Angew. Math. 364, 35–50 (1986).

    MATH  MathSciNet  Google Scholar 

  70. C. Okonek and A. Van de Ven, Vector bundles and geometry: Some open problems, Math. Gottingensis 17 (1988).

    Google Scholar 

  71. G. Ottaviani, Some extensions of Horrocks criterion to vector bundles on Grassmannians and quadrics, Ann. Mat. 155, 317–341 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  72. P. Pragacz and M. Szurek, Bounds of Chern classes of semistable vector bundles, Math. Gottingensis, Schriften. Sonderforsch. Geometrie und Analysis, Heft 22 (1990).

    Google Scholar 

  73. E. Previato, Generalized Weierstrass p-funetions and KP flows in affine space, Comment. Math. Helv. 62, 292–310 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  74. E. Previato and G. Wilson, Vector bundles over curves and solutions of the KP equations, Proc. Symp. Pure Math., Vol. 49, pp. 553–559 (L. Ehrenpreis and R. C. Gunning, eds.), American Mathematical Society, Providence, RI (1989).

    Google Scholar 

  75. E. Previato and G. Wilson, Differential operators and rank two bundles over elliptic curves, preprint (1990).

    Google Scholar 

  76. S. Ramanan, Orthogonal and spin bundles over hyperelliptic curves, in V. K. Patodi Memorial Volume, Tata Institute, Bombay (1981).

    Google Scholar 

  77. A. Prabhakar Rao, A family of vector bundles on P3, in Space Curves, Lecture Notes in Math., Vol. 1266, pp. 208–231, Springer-Verlag, Berlin and New York (1987).

    Chapter  Google Scholar 

  78. M. Reid, The complete intersection of two or more quadrics, Dissertation Trinity College, Cambridge (1971).

    Google Scholar 

  79. A. N. Rudakov, Markov numbers and exceptional bundles on P2, Izv. Akad. Nauk SSSR Ser. Mat. 52, 100–112 (1988) (Russian); English transi, in Math. USSR Izv. 32 (1989).

    Google Scholar 

  80. A. N. Rudakov, Exceptional vector bundles on a quadric, Izv. Akad. Nauk SSSR Ser. Math. 52 (1988) (Russian); English transi, in Math. USSR Izv. 32, 115–138 (1989).

    Google Scholar 

  81. U. Schafft, Nichtsepariertheit instabiler Rang-2 Vector bundel auf P2, J. Reine Angew. Math. 338, 136–143 (1983).

    MATH  MathSciNet  Google Scholar 

  82. R. Schilling, Generalization of the Neumann system—A curve theoretical approach. I, II, Comm. Pure Appl. Math. 40, 455–522 (1987);

    Article  MATH  MathSciNet  Google Scholar 

  83. R. Schilling, Generalization of the Neumann system—A curve theoretical approach. I, II, Comm. Pure Appl. Math. 42, 409–442 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  84. M. Schneider, Cherklassen semi-stabiler Vectorraumbundel vom Rang 3 auf dem komplex-projektiven Raum, J. Reine Angew. Math. 323, 211–220 (1982).

    Google Scholar 

  85. M. Schneider, Vector bundles and submanifolds of projective space : Nine open problems, in Algebraic Geometry (Bowdoin 1985), Proc. Symp. Pure Math., Vol. 46, Part 2, pp. 101 – 107, American Mathematical Society, Providence, RI (1987).

    Google Scholar 

  86. M. Schneider, Vector bundles and low-codimensional submanifolds of projective space: A problem list, in Topics in Algebra, Banach Center Publications, Vol. 26, Part 2, pp. 209–222, PWN-Polish Scientific Publishers, Warsaw (1990).

    Google Scholar 

  87. R. L. E. Schwarzenberger, Vector bundles on algebraic surfaces, Proc. London Math. Soc. (3)11, 601–622 (1961).

    Article  MATH  MathSciNet  Google Scholar 

  88. H. Spindler and G. Trautmann, Special instanton bundles on p2n+1, their geometry and their moduli, Math. Ann. 286, 559–592 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  89. S. A. Stromme, Deforming vector bundles on the projective plane, Math. Ann. 263, 385–397 (1983).

    Article  MathSciNet  Google Scholar 

  90. S. I. Svinolupov, V. Sokolov, and R. Yamilov, On Backlund transformations for integrable evolution equation, Soviet Math. Dokl. 28, 165–168 (1983).

    MATH  Google Scholar 

  91. M. Szurek and J. A. Wisniewski, Fano bundles on P3 and Q3, Pacific J. Math. 141, 197–208 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  92. M. Szurek and J. A. Wisniewski, On Fano manifolds which are Pk-bundles over P2, Nagoya Math. J. 120, 89–101 (1990).

    MATH  MathSciNet  Google Scholar 

  93. M. Szurek and J. A. Wisniewski, Del Pezzo surfaces as conic bundles, preprint.

    Google Scholar 

  94. M. Toma, Une classe de fibres vectoriels holomorphes sur les 2-tore complexe, C. R. Acad. Sci. Paris 311, 257–258 (1990).

    MATH  MathSciNet  Google Scholar 

  95. A. Van de Ven, Twenty years of classifying algebraic vector bundles, in Journées de Géométrie Algébrique, Sijitoff and Noordhoff (1980).

    Google Scholar 

  96. H. Vollinger, Moduli von Kernbundeln auf Pat(C), dissertation, Kaiserslautern (1988).

    Google Scholar 

  97. C. Weibel, K-theory and analytic isomorphisms, Invent. Math. 61, 177–197 (1980).

    Article  MATH  MathSciNet  Google Scholar 

  98. L. Weng, Non-degeneracy theorem: a note for generic smooth theorem, Max-Planck-Institut, preprint 90–39.

    Google Scholar 

  99. G. Wilson, On the quasi-Hamiltonian formalism of the KdV equation, Phys. Lett. A 132, 445–450 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  100. J. A. Wisniewski, Ruled Fano 4-folds of index 2, Proc. Am. Math. Soc. 105, 55–61 (1988).

    Article  MathSciNet  Google Scholar 

  101. K. Zuo, Regular 2-forms on the moduli space of rank 2 stable bundles on an algebraic surface.

    Google Scholar 

  102. K. Zuo, Smoothness of the moduli of rank two stable bundles over an algebraic surface.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Trento, 38050, Povo TN, Italy

    Edoardo Ballico

Authors
  1. Edoardo Ballico
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Università di Firenze, Florence, Italy

    Vincenzo Ancona

  2. Università di Roma “La Sapienza”, Rome, Italy

    Alessandro Silva

Rights and permissions

Reprints and permissions

Copyright information

© 1993 Springer Science+Business Media New York

About this chapter

Cite this chapter

Ballico, E. (1993). A Problem List on Vector Bundles. In: Ancona, V., Silva, A. (eds) Complex Analysis and Geometry. The University Series in Mathematics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-9771-8_17

Download citation

  • DOI: https://doi.org/10.1007/978-1-4757-9771-8_17

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4757-9773-2

  • Online ISBN: 978-1-4757-9771-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation