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We study the Albanese image of a compact K\"ahler manifold whose geometricgenus is one. We prove that if the Albanese map is not surjective, then themanifold maps surjectively onto an ample divisor in some abelian variety, andin many cases the ample divisor is a theta divisor. With a further naturalassumption on the topology of the manifold, we prove that the manifold is analgebraic fiber space over a genus two curve. Finally we apply these results tostudy the geometry of a compact K\"ahler manifold which has the same Hodgenumbers as those of an abelian variety of the same dimension.

Irregular varieites with geometric genus one, theta divisors, and fake tori