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

TL;DR

This paper proves that very general hypersurfaces of bidegree (k,k) in P^n x P^n exhibit asymptotic purity, meaning most asymptotic cohomological functions vanish, which supports a conjecture by Bogomolov.

Contribution

It establishes asymptotic purity for a broad class of hypersurfaces, advancing understanding of positivity and cohomological properties in algebraic geometry.

Findings

Very general hypersurfaces of bidegree (k,k) are asymptotically pure.

Supports Bogomolov's conjecture on asymptotic cohomological behavior.

Provides conditions under which asymptotic purity holds.

Abstract

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