  This article is an expanded version of the talk given by the first author at
the conference "Exponential sums over finite fields and applications" (ETH,
Z\"urich, November, 2010). We state some conjectures on archimedian and
$p$-adic estimates for multiplicative character sums over smooth projective
varieties. We also review some of the results of J. Dollarhide, which formed
the basis for these conjectures. Applying his results, we prove one of the
conjectures when the smooth projective variety is ${\mathbb P}^n$ itself.
