What is the definition of a holomorphic $\mathbb{C}^\ast$-action on a complex manifold $M$?
My understanding is that it is a group action of the Lie group $\mathbb{C}^\ast$ (the multiplicative group of complex numbers) on $M$ such that the map $ f: \mathbb{C}^\ast \times M \to M $ is holomorphic, which means that the map $ f $ from the complex manifold $ \mathbb{C}^{\ast} \times M $ (the Cartesian product of two complex manifolds is again a complex manifold) to $ M $ is holomorphic. That is, for all $ p \in \mathbb{C}^\ast \times M $ there is a chart $ (U, \phi) $ on $ \mathbb{C}^\ast \times M $ with $ p \in U $ and a chart $ (V, \psi) $ on $ M $ with $ \phi(p) \in V $ such that $ f(U) \subseteq V $ and the map $$ \phi(U) \stackrel{\phi^{-1}}\longrightarrow U \stackrel{f}{\to} V \stackrel{\psi}{\to}\mathbb{C}^n $$ is holomorphic (it is a map from $\phi(U) \subseteq \mathbb{C}^{n+1}$ to $ \mathbb{C}^{n} $, where $ n $ is the dimension of $ M $, so $ n+1 $ is the dimension of $ \mathbb{C}^{\ast} \times M $).
Is my definition correct?