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For a complex irreducible projective variety, the volume function and thehigher asymptotic cohomological functions have proven to be useful inunderstanding the positivity of divisors as well as other geometric propertiesof the variety. In this paper, we study the vanishing properties of thesefunctions on specific hypersurfaces of P^n \times P^n. In particular, we showthat very general hypersurfaces of bidegree (k,k) obey a very strong vanishingproperty, which we define as asymptotic purity: at most one asymptoticcohomological function is nonzero for each divisor. This provides evidence fora conjecture of Bogomolov and also suggests some general conditions forasymptotic purity.

Asymptotic Purity for Very General Hypersurfaces of P^n x P^n of Bidegree (k,k)