Surfaces generating the even primal cohomology of an abelian fivefold

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 $$E_6$$ .