Smooth metrics on jet bundles and applications

TL;DR

The paper constructs smooth hermitian metrics on jet bundles of complex manifolds, computes their curvature, and applies these results to prove the existence of global invariant jet differentials on certain surfaces without relying on strong vanishing theorems.

Contribution

It introduces a natural smooth hermitian metric on jet bundles, computes its curvature recursively, and applies this to establish the existence of invariant jet differentials on minimal surfaces of general type.

Findings

Curvature of the constructed metrics depends asymptotically on the base curvature and fibration structure.

Explicit curvature formulas are derived for surfaces with the tangent bundle.

New proof of existence of global invariant jet differentials on certain surfaces.

Abstract

Following a suggestion made by J.-P. Demailly, for each $k\ge 1$, we endow, by an induction process, the $k$-th (anti)tautological line bundle $\mathcal O_{X_k}(1)$ 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 $X_k\to X$. When $X$ is a surface and $V=T_X$, 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 of Bogomolov.