We define locally trivial quantum vector bundles (QVB) and construct such QVB associated to locally trivial quantum principal fibre bundles. The construction is quite analogous to the classical construction of associated bundles. A covering of such bundles is induced from the covering of the subalgebra of coinvariant elements of the principal bundle. There exists a differential structure on the...

The purpose of this paper is twofold. The rst purpose is to give a simple treatment of the basic theory of atomic sections and their zero currents or divisors. This theory (see HS,HL1]) places the classical theory of characteristic classes on a new and more general analytic footing based on de Rham currents and geometric measure theory. The second purpose is to discuss an instance of the Cheege...

Let p : X → S be a smooth Kähler fibration and E → X a Hermitian holomorphic vector bundle. As motivated by the work of Berdtsson([Bern09]), by using basic Hodge theory, we derive several general curvature formulas for the direct image p∗(KX/S ⊗E) for general Hermitian holomorphic vector bundle E in a very simple way. A straightforward application is that, if the Hermitian vector bundle E is Na...

We consider filtered holomorphic vector bundles on a compact Riemann surface X equipped with a holomorphic connection satisfying a certain transversality condition with respect to the filtration. If Q is a stable vector bundle of rank r and degree (1−genus(X))nr , then any holomorphic connection on the jet bundle Jn(Q) satisfies this transversality condition for the natural filtration of Jn(Q) ...

We recall the basic theory of double vector bundles and the canonical pairing of their duals introduced by the author and by Konieczna and Urbański. We then show that the relationship between a double vector bundle and its two duals can be understood simply in terms of an associated cotangent triple vector bundle structure. In particular we show that the dihedral group of the triangle acts on t...