Green's formula with ${\text{bbc}}^{\ast }$-action and Caldero-Keller's formula for cluster algebras

It is known that Green’s formula over finite fields gives rise to the comultiplications of Ringel-Hall algebras and quantum groups (see\cite{Green}, also see \cite{Lusztig}). In this paper, we deduce the projective version of Green’s formula in a geometric way. Then following the method of Hubery in \cite{Hubery2005}, we apply this formula to proving Caldero-Keller’s multiplication formula for acyclic cluster algebras of arbitrary type.