UC San Diego

Given a general abelian fivefold A and a symmetric principal polarisation Θ ⊂ A, the primal cohomology of Θ is the part which is not inherited from A. We compute numerical invariants of the primal cohomology lattice, and construct surfaces inside Θ whose classes span the subspace fixed by -1 (with rational coefficients). This gives a constructive proof of the rational Hodge conjecture for Θ. The sublattice generated by the surfaces is (up to a factor of 2) isometric to the root lattice E6.