  Starting from a manifestly Lorentz- and diffeomorphism-invariant classical
action we perform a perturbative derivation of the gravitational anomalies for
chiral bosons in 4n+2 dimensions. The manifest classical invariance is achieved
using a newly developed method based on a scalar auxiliary field and two new
bosonic local symmetries. The resulting anomalies coincide with the ones
predicted by the index theorem. In the two-dimensional case, moreover, we
perform an exact covariant computation of the effective action for a chiral
boson (a scalar) which is seen to coincide with the effective action for a
two-dimensional complex Weyl-fermion. All these results support the quantum
reliability of the new, at the classical level manifestly invariant, method.
