I am currently struggling my way through Claude LeBrun's paper Complete Ricci-flat metrics on $\mathbb{C}^n$ need not be flat and have a few questions that I would appreciate some help with.
Let $V : = 1 + 1/2r : \mathbb{R}^3 - \{ 0 \} \to \mathbb{R}^+$, and let $\pi : M_0 \to \mathbb{R}^3 - \{ 0 \}$ be the $\mathbb{S}^1$-bundle of Chern class $-1$. We equip this circle bundle with a connection of curvature $F : = \star dV$ and let $\omega$ denote the connection form of this connection. If $g_{\mathbb{R}^3}$ denotes the standard Euclidean metric, then $$g = V \pi^{\ast} g_{\mathbb{R}^3} + V^{-1} \omega^2$$ gives us a Riemannian metric on $M_0$ called the Taub-NUT metric.
Claim: The above metric is Kähler with respect to the complex structure defined by sending the horizontal lift of a unit vector (viewed as a constant vector field on $\mathbb{R}^3$) to the vertical Killing field while preserving $g$.
More precisely, let us take our unit vector to be $\partial/\partial x$, and take some trivialisation of $\pi : M_0 \to \mathbb{R}^3$ with vertical coordinate $t$, so that $\omega = dt + \vartheta$ for some $1$-form $\vartheta$ n $\mathbb{R}^3$ satisfying $d\vartheta = \star dV$ (this represents the connection in this gauge). Then our complex structure $J$ is given on 1-forms by $J(dx) = V^{-1}(dt + \vartheta)$ and $J(dy) = dz$. Note that this tells us that $J(dt + \vartheta) = -Vdx$ and $J(dz) = -dy$.
Q1: The complex structure is typically written down as an endomorphism of the tangent bundle. Is it possible to explicitly write down the representation of the above complex structure as a map $TM \to TM$?
Q2: Can someone help elucidate why the complex structure is defined in this way?
Finally, later in the paper, the author considers the vector field $$\xi = -\frac{i}{2}(\partial_t - i J \partial_t) = \frac{1}{2}(V^{-1}\hat{\partial}_x - i \partial_t),$$ where $\hat{\partial}_x = \partial_x - \vartheta(\partial_x)\partial_t$ is the horizontal lift of $\partial_x$.
Q3: The author claims that $\partial_t$ preserves both the metric and the complex structure, i.e., $L_{\partial_t}(J) = L_{\partial_t}(g) =0$ where $L_{\partial_t}$ denotes the Lie derivative with respect to $\partial_t$, but I cannot see why this is the case.
Remark: I understand that these questions are rather elementary, but I've been looking at this for a few days and have decided I need some help. Thanks in advance.
Remark: Please let me know if you are unable to find the paper, it was rather difficult for me to find.