How does a choice of linearisation of a line bundle fix a moment map?

Question

Let $G \times M \to M$ be a weak Hamiltonian action of a Lie group on a Kahler manifold. Suppose we fix a lift/linearisation of the action of $G$ to an ample line bundle $L \to M$.

Apparently this fixes a canonical choice of moment map. Can someone explain this to me?

I don't know whether this is correct, but we have an embedding in $\mathbb{P}^N$ given by $L$. Does the linearisation induce an action on $\mathbb{P}^N$, where we do have a canonical choice of moment map?

Answer 1

Check the notes of Victoria Hoskins "SYMPLECTIC QUOTIENTS: MOMENT MAPS, SYMPLECTIC REDUCTION AND THE MARSDEN-WEINSTEIN-MEYER THEOREM"