  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.
