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Hermitian-Yang-Mills approach to the conjecture of Griffiths on the positivity of ample vector bundles. (English. Russian original) Zbl 1464.32028

Sb. Math. 212, No. 3, 305-318 (2021); translation from Mat. Sb. 212, No. 3, 39-53 (2021).
Summary: 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.

MSC:

32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
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