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 hypersurfaces of ℙn × ℙn. 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 the truth of a conjecture of Bogomolov and also suggests some general conditions for asymptotic purity.
@article{bwmeta1.element.doi-10_2478_s11533-011-0126-8,
author = {Michael Burr},
title = {Asymptotic purity for very general hypersurfaces of Pn x Pn of bidegree (k, k)},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {530-542},
zbl = {1273.14042},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0126-8}
}
Michael Burr. Asymptotic purity for very general hypersurfaces of ℙn × ℙn of bidegree (k, k). Open Mathematics, Tome 10 (2012) pp. 530-542. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0126-8/
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