Hermitian-Yang-Mills approach to the conjecture of Griffiths on the positivity of ample vector bundles

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Аннотация

Given a vector bundle of arbitrary rank with ample determinant line bundle on a projective manifold, we propose a new elliptic system of differential equations of Hermitian-Yang-Mills type for the curvature tensor. The system is designed so that solutions provide Hermitian metrics with positive curvature in the sense of Griffiths — and even in the dual Nakano sense. As a consequence, if an existence result could be obtained for every ample vector bundle, the Griffiths conjecture on the equivalence between ampleness and positivity of vector bundles would be settled. Bibliography: 15 titles.

Авторлар туралы

Jean-Pierre Demailly

Institut Fourier, UFR de Mathématiques

Email: jean-pierre.demailly@univ-grenoble-alpes.fr
PhD

Әдебиет тізімі

  1. B. Berndtsson, “Curvature of vector bundles associated to holomorphic fibrations”, Ann. of Math. (2), 169:2 (2009), 531–560
  2. F. Campana, H. Flenner, “A characterization of ample vector bundles on a curve”, Math. Ann., 287:4 (1990), 571–575
  3. J.-P. Demailly, H. Skoda, “Relations entre les notions de positivites de P. A. Griffiths et de S. Nakano pour les fibres vectoriels”, Seminaire Pierre Lelong–Henri Skoda (Analyse). Annees 1978/79, Lecture Notes in Math., 822, Springer, Berlin, 1980, 304–309
  4. S. K. Donaldson, “Anti self-dual Yang–Mills connections over complex algebraic surfaces and stable vector bundles”, Proc. London Math. Soc. (3), 50:1 (1985), 1–26
  5. P. A. Griffiths, “Hermitian differential geometry, Chern classes and positive vector bundles”, Global analysis, Papers in honor of K. Kodaira, Univ. Tokyo Press, Tokyo, 1969, 181–251
  6. K. Kodaira, “On Kähler varieties of restricted type (an intrinsic characterization of algebraic varieties)”, Ann. of Math. (2), 60 (1954), 28–48
  7. C. Mourougane, S. Takayama, “Hodge metrics and positivity of direct images”, J. Reine Angew. Math., 2007:606 (2007), 167–178
  8. S. Nakano, “On complex analytic vector bundles”, J. Math. Soc. Japan, 7 (1955), 1–12
  9. M. S. Narasimhan, C. S. Seshadri, “Stable and unitary vector bundles on a compact Riemann surface”, Ann. of Math. (2), 82:3 (1965), 540–567
  10. P. Naumann, An approach to Griffiths conjecture
  11. V. P. Pingali, “A vector bundle version of the Monge–Ampère equation”, Adv. Math., 360 (2020), 106921, 40 pp.
  12. V. P. Pingali, “A note on Demailly's approach towards a conjecture of Griffiths”, C. R. Math. Acad. Sci. Paris, 2021 (to appear)
  13. K. Uhlenbeck, S. T. Yau, “On the existence of Hermitian–Yang–Mills connections in stable vector bundles”, Comm. Pure Appl. Math., 39:S, suppl. (1986), 257–293
  14. H. Umemura, “Some results in the theory of vector bundles”, Nagoya Math. J., 52 (1973), 97–128
  15. Shing-Tung Yau, “On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. I”, Comm. Pure Appl. Math., 31:3 (1978), 339–411

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