  We consider a short sequence of hermitian vector bundles on some arithmetic
variety. Assuming that this sequence is exact on the generic fiber we prove
that the alternated sum of the arithmetic Chern characters of these bundles is
the sum of two terms, namely the secondary Bott Chern character class of the
sequence and its Chern character with supports on the finite fibers. Next, we
compute these classes in the situation encountered by the second author when
proving a "Kodaira vanishing theorem" for arithmetic surfaces.
