Surfaces generating the even primal cohomology of an abelian fivefold

TL;DR

This paper constructs specific surfaces within a general abelian fivefold's principal polarization to generate the algebraic part of its middle cohomology, providing new insights into the Hodge conjecture and lattice structures.

Contribution

It introduces a novel method to generate the algebraic part of the middle cohomology of a general abelian fivefold's polarization and proves the Hodge conjecture in this context.

Findings

Surfaces generate the algebraic part of H^4(Θ, Q).

Intersection pairing between these surfaces is explicitly determined.

H^4(Θ, Q) contains a copy of the E_6 root lattice.

Abstract

Given a very general abelian fivefold $A$ and a principal polarization $\Theta \subset A$, we construct surfaces generating the algebraic part of the middle cohomology $H^4(\Theta, {\mathbb Q})$, and determine the intersection pairing between these surfaces. In particular, we obtain a new proof of the Hodge conjecture for $H^4(\Theta, {\mathbb Q})$ and show that it contains a copy of the root lattice of $E_6$.