I am now reading Feix's paper *Hyperkahler metrics on cotangent bundles*
and I have a technical question to ask.

In his paper, for an analytic Kähler manifold $(X,J,\omega)$, Feix considered its complexification $X^c$ which in my understanding can be thought of as a neighborhood of the diagonal in $X\times\bar{X}$, where $\bar{X}$ is the complex manifold $X$ with complex structure $-J$. One can extend $\omega$ analytically to a holomorphic symplectic form $\omega^c$ on $X^c$. This $\omega^c$ determines two natural holomorphic Lagrangian foliations $L_+$ and $L_-$. Let $z_i$ and $z'_j$ be local holomorphic coordinates of $X$ and $\bar{X}$ respectively, then the leaves of $L_+$ and $L_-$ are given by $z_i\equiv const$ and $z'_j\equiv const$.

As the diagonal intersects each leaf at exactly one point, one may identify the space of leaves of $L_+$ with $X$ and the space of leaves of $L_-$ with $\bar{X}$ respectively.

From now on let us focus on $L_+$ exclusively. By shrinking $X^c$ if necessary, one may assume that $\Lambda_x$ is simply connected for any $x\in X=$ "space of leaves of $L_+$", where $\Lambda_x$ is the leaf corresponding to $x$. As a consequence of Lagrangian foliation, each $\Lambda_x$ has a natural affine structure, so it makes sense to write $V_x$ to be the space of affine functions on $\Lambda_x$. By 1-connectedness of $\Lambda_x$, each $V_x$ is a vector space of complex dimension $n+1$, where $n$ is the complex dimension of $X$. These $V_x$ patch up to a complex vector bundle $V\to X$.

Feix claims without further explanation that this bundle is holomorphic. I really would like to know the description of the holomorphic structure here. It occurs to me that the most natural frames one can think of actually have anti-holomorphic transition functions as follows:

Let $z_i,z'_j$ be local coordinates for $X^c$, the leaves of $L_+$ are $z_i\equiv const$. Fix a leaf $\Lambda_x$, it intersects the diagonal at the point whose coordinate is $z=x,z'=\bar{x}$. One can find parallel 1-forms $\theta_i$ on $\Lambda_x$ by parallel transport with respect to the flat connection with initial value specified by $\theta_i|_{(x,\bar{x})}=\textrm{d}z'_i|_{(x,\bar{x})}$. These $\theta_i$ must be a closed form and their primitives $f_i$ along with the constant function 1 form a basis of $V_x$. However, if you work with this particular frame, then under holomorphic coordinate change of $X$, the transition matrix of these frame depends anti-holomorphically on $X$.

Surely one can switch $L_+$ and $L_-$ to solve the holomorphicity problem here. But I think a bigger trouble is then introduced since in that case the map $\phi$ defined by Feix loses its holomorphicity.

Thank you!