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A special class of symplectic manifolds is those satisfying the HardLefschetz condition. The algebraic property of $\mathfrak{sl}_2(\mathbbC)$-module gives a natural constraint on the Betti numbers. However, such aconstraint is only necessary, and it is not so clear what condition on theBetti numbers is equivalent to the Hard Lefschetz condition. In this paper, wegive some relations between the Betti numbers and the variant Hodge numbers onthe Hard Lefschetz condition. In general, we give a sufficient condition onBetti numbers and variant Hodge numbers for any almost K\"ahler manifoldsatisfying the Hard Lefschetz condition. In particular, we offer an equivalentcondition on the first Betti number for compact symplectic $4$-manifoldssatisfying the Hard Lefschetz condition.

Betti numbers and compactible almost complex structures on the Hard Lefschetz condition