Smooth metrics on jet bundles and applications

Following a suggestion made by J.-P. Demailly, for each k≥1, we endow, by an induction process, the k-th (anti)tautological line bundle O_X_k(1) of an arbitrary complex directed manifold (X,V) with a natural smooth Hermitian metric. Then, we compute recursively the Chern curvature form for this metric, and we show that it depends (asymptotically---in a sense to be specified later) only on the curvature of V and on the structure of the fibration Xk→X. When X is a surface and V=TX, we give explicit formulae to write down the above curvature as a product of matrices. As an application, we obtain a new proof of the existence of global invariant jet differentials vanishing on an ample divisor, for X a minimal surface of general type whose Chern classes satisfy certain inequalities, without using a strong vanishing theorem [1] of Bogomolov.