Constraints on the Action of Positive Correspondences on Cohomology

Joseph Knight, Purdue University

Abstract

Let X be a smooth projective variety of dimension n over an algebraically closed field K. We investigate constraints that the positivity of a correspondence (i.e., an algebraic n-cycle on X × X, usually considered modulo homological or numerical equivalence) imposes on its action on `-adic (or, over C, its singular) cohomology. For example, conjectures of DebarreJiang-Voisin imply nontrivial restrictions on this action whenever (in our terminology) a strictly pseudoeffective homological correspondence lies in the kernel of the pushforward by one of the projections X × X → X. We prove some new cases of these conjectures for correspondences on hyperelliptic varieties. In another direction, if the cycle class ∆X of the diagonal subvariety satisfies a certain positivity condition (which holds, for example, if X has nef tangent bundle), and A is a correspondence that is pseudoeffective (modulo numerical equivalence), then we show that the spectral radius of A (with respect to its action on cohomology) is attained on the even-degree cohomology. Under some further conditions (e.g. if K = Fq or X is an abelian variety), the spectral radius is in fact attained on the part of cohomology generated by iterates of A acting on powers of an ample divisor class. The key to these results is the existence and properties of a “good” cone C(X × X) of positive correspondences that is preserved under composition. If ∆X is an interior point of C(X ×X), then we show that X is fake projective (i.e. that X has the same `-adic betti numbers as Pn ). If C(X × X) is finitely-generated, then we show (under a few further conditions) that X is “fake rational homogeneous” in a certain sense (in particular, the cohomology of Xis entirely algebraic).

Degree

Ph.D.

Advisors

Patel, Purdue University.

Subject Area

Mathematics

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