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The Demailly system for a direct sum of ample line bundles on Riemann surfaces

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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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  1. The author thanks J.-P. Demailly for this observation, which was already mentioned by Liu–Sun–Yang [5].


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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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Correspondence to Vamsi Pritham Pingali.

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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).

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