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