Greens formula with $bbc^{*}$-action and Caldero-Kellers formula for cluster algebras

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