Contact Ozsváth–Szabó invariants and Giroux torsion

In this paper we prove a vanishing theorem for the contact Ozsvath--Szabo invariants of certain contact 3--manifolds having positive Giroux torsion. We use this result to establish similar vanishing results for contact structures with underlying 3--manifolds admitting either a torus fibration over the circle or a Seifert fibration over an orientable base. We also show -- using standard techniques from contact topology -- that if a contact 3--manifold (Y,\xi) has positive Giroux torsion then there exists a Stein cobordism from (Y,\xi) to a contact 3--manifold (Y,\xi') such that (Y,\xi) is obtained from (Y,\xi') by a Lutz modification.