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.


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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