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We first study a new family of graded quiver varieties together with a new$t$-deformation of the associated Grothendieck rings. This provides thegeometric foundations for a joint paper by Yoshiyuki Kimura and the author.  We further generalize the result of that paper to any acyclic quantum clusteralgebra with arbitrary nondegenerate coefficients. In particular, we obtain thegeneric basis, the dual PBW basis, and the dual canonical basis. The methodconsists in a correction technique, which works for general quantum clusteralgebras.

$t$-Analog of $q$-Characters, Bases of Quantum Cluster Algebras, and a Correction Technique