This paper discusses the problem of whether it is possible
to annihilate elements of local cohomology modules by elements
of arbitrarily small order under a fixed valuation. We first
discuss the general problem and its relationship to the Direct
Summand Conjecture, and next present two concrete examples
where annihilators with small order are shown to exist. We
then prove a more general theorem, where the existence of such
annihilators is established in some cases using results on
abelian varieties and the Albanese map.