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

Jean-Pierre Demailly (1)

1 IF - Institut Fourier

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.

Mots clés: Ample vector bundle; Hermitian metric; Griffths positivity; Nakano positivity; Hermitian-Yang-Mills equation; Monge-Ampère equation; elliptic operator; elliptic operator. MSC Classification 2020. 32J25; 53C07

Domaines: Variables complexes [math.CV]; Géométrie différentielle [math.DG]

Jean-Pierre Demailly. Hermitian-Yang-Mills approach to the conjecture of Griffiths on the positivity of ample vector bundles. 2020. ⟨hal-02468559v4⟩