Suppose we have a smooth complex projective variety $X$ with a holomorphic action of a complex reductive algebraic group $G$ (i.e. the map $X \times G \rightarrow X$ is holomorphic). Suppose that the group action can be linearised i.e. the group action lifts to a holomorphic action on the space of some very ample line bundle (by bundle isomorphisms). Then we get an induced action of $G$ on the ambient projective space of some embedding of $X$ (since we have $G$-representation $H^{0}(X,L)$ ). Question: Is this group action holomorphic?
Stated another way, the linearization gives a natural injective group homomorphism $Aut(X) \rightarrow Aut(\mathbb{P}^{n}) = PGL(n+1,\mathbb{C})$, is this map holomorphic? (Note that there is a canonical complex lie group structure on $Aut(X)$, so the question is well-posed). I have looked in several sources, for example the book of Kobayashi on transformation groups, but they never say this explicitly (I have a feeling it is so obvious that they don't bother to state this, but I want to check).