Abstract
We prove that a system of equations introduced by Demailly (to attack a conjecture of Griffiths) has a smooth solution for a direct sum of ample line bundles on a Riemann surface. We also reduce the problem for general vector bundles to an a priori estimate using Leray–Schauder degree theory.
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Notes
The author thanks J.-P. Demailly for this observation, which was already mentioned by Liu–Sun–Yang [5].
References
Berndtsson, B.: Curvature of vector bundles associated to holomorphic fibrations. Ann. Math. 169, 531–560 (2009)
Campana, F., Flenner, H.: A characterization of ample vector bundles on a curve. Math. Ann. 287, 571–575 (1990)
Demailly, J.-P.: Hermitian–Yang–Mills approach to the conjecture of Griffiths on the positivity of ample vector bundles. Sbornik Math. 212(3), 305 (2021)
Griffiths, P.A.: Hermitian differential geometry, Chern classes and positive vector bundles. In: Global analysis (papers in honor of K. Kodaira), University of Tokyo Press & Princeton University Press, pp. 185–251 (1969)
Liu, K., Sun, X., Yang, X.: Positivity and vanishing theorems for ample vector bundles. J. Alg. Geom. 22(2), 303–331 (2013)
Mandal, A.: The Demailly systems with the Vortex ansatz, arXiv preprint, arXiv:2301.09076
Mourougane, C., Takayama, S.: Hodge metrics and positivity of direct images. J. Reine Angew. Math. 606, 167–178 (2007)
Pingali, V.: Representability of Chern–Weil forms. Math. Z. 288, 629–641 (2018)
Pingali, V.: A note on Demailly’s approach towards a conjecture of Griffiths. C. R. Math. Acad. Sci. Paris 359(4), 501–503 (2021)
Uhlenbeck, K., Yau, S.T.: On the existence of Hermitian–Yang–Mills connections in stable vector bundles. Commun. Pure App. Math. 39(S1), S257–S293 (1986)
Umemura, H.: Some results in the theory of vector bundles. Nagoya Math. J. 52, 97–128 (1973)
Acknowledgements
This work is partially supported by grant F.510/25/CAS-II/2018(SAP-I) from UGC (Govt. of India), and a MATRICS grant MTR/2020/000100 from SERB (Govt. of India). The author is sincerely grateful to J.-P Demailly for his kind support, encouragement, and useful mathematical discussions. Demailly (may he rest in peace) will always be a source of inspiration to the author. We also thank Ved Datar, Richard Wentworth, Sandeep Kunnath, and Swarnendu Sil for fruitful discussions. Lastly, thanks is in order to the anonymous referee for useful suggestions.
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Communicated by Richard M. Schoen.
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Pingali, V.P. The Demailly system for a direct sum of ample line bundles on Riemann surfaces. Calc. Var. 62, 172 (2023). https://doi.org/10.1007/s00526-023-02517-3
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DOI: https://doi.org/10.1007/s00526-023-02517-3