Filtering the Heegaard Floer contact invariant

We define an invariant of contact structures in dimension three from Heegaard Floer homology. This invariant takes values in the set Z_≥0 ∪{∞}. It is zero for overtwisted contact structures, ∞ for Stein-fillable contact structures, nondecreasing under Leg-endrian surgery, and computable from any supporting open book decomposition. As an application, we give an easily computable obstruction to Stein-fillability on closed contact 3–manifolds with nonvanishing Ozsváth–Szabó contact class.