  We introduce exponential complexes of sheaves on manifolds. They are
resolutions of the (Tate twisted) constant sheaves of the rational numbers,
generalising the short exact exponential sequence. There are canonical maps
from the exponential complexes to the de Rham complex. Using this, we introduce
new complexes calculating the rational Deligne cohomology. We call them
exponential Deligne complexes. Their advantage is that, at least at the generic
point of a complex variety, one can define Beilinson's regulator map to the
rational Deligne cohomology on the level of complexes. Namely, we define period
morphisms. We use them to produce homomorphisms from motivic complexes to the
exponential Deligne complexes at the generic point. Combining this with the
construction of Chern classes with coefficients in the bigrassmannian
complexes, we get, for the weights up to four, local explicit formulas for the
Chern classes in the rational Deligne cohomology via polylogarithms.
