I am trying to understand this passage of the paper "APPLICATIONS OF THE HOLOMORPHIC LEFSCHETZ FORMULA" from Kosniowski
1) Just to make sure I understand the notion correctly: Let $n$ be the complex dimension of the manifold $X$. A zero point $P$ of a holomorphic vector field means that for a frame $\partial / \partial z_1, \ldots, \partial / \partial z_n$ of the tangent space around that point the vector field has the form $\sum f_i \partial / \partial z_i$, where $f_i$'s are holomorphic $n$-variable functions with zero at $P$ but there is at least one $f_i$ with nonzero gradient at $P$.
2) How does $A$ induce the linear endomorphism $L_p(A)$ at $P$? My guess is for a vector $v \in T_PX$ we take a (local) vector field (even smooth?) $V$ with $v$ at $p$, then the Lie bracket gives $L(A)(V) = AV - VA$ a vector field, s.t. when we apply on a function $g$ and look at point $P$ we get: $$AV(g) - VA(g) = V(A(g))$$ because $A$ is zero at $p$ which means the endomorphism $L_P(A)$ applied on $v$ is just $VA$ at point $P$. This is clearly independent on the choice of $V$.
3) The exponential map as I understand $g_t = exp(tA) \colon X \rightarrow X$ is defined such that for each point $x \in X$ we define a curve $\gamma \colon [0,1] \rightarrow X$ with $\gamma(0) = x$ and the tangent at each point of the curve coincides with $A$, then $exp(tA)(x) = \gamma(t)$. The fixed points of $g_t$ are the zero set of $A$ because when the tangent is zero the curve stays at that point.
The map $dg_t$ at a fixed point $P$ is a map $T_PX \rightarrow T_PX$ s.t. for $v \in T_PX$ we have $dg_t(v)(f) = v(f \circ g_t)$. Why does $dg_t$ at $P$ have the form $exp(tL_p(A))$. It seems to be a standard fact from differential geometry but I cannot find it anywhere.
I would appreciate an explanation, a confirmation that what I wrote makes sense or reference to relevant literature. Thank you.
