This question/reference request is related to the books Compact manifolds with special holonomy, Riemannian holonomy groups and calibrated geometry and the paper Manifolds with many complex structures all by D. Joyce.
In all of this there is the fallowing notation. Let $(M,J)$ be a complex manifold of real dimension $2m$. Fix some (real) coordinate chart $(U,x_1,...,x_{2m})$. Suppose we have a tensor $t=t^{a...}_{b...}$ i.e. $t=\sum_{a,...,b,...}\partial_{x_1} \otimes ...\otimes dx_b\otimes ...$ . Now he defines (tensors?) $$t^{\alpha ...}_{...}=\frac{1}{2}(t^{a ...}_{...}-i J^a_ct^{c ...}_{...}), \:\:t^{\overline{\alpha} ...}_{...}=\frac{1}{2}(t^{a ...}_{...}+i J^a_ct^{c ...}_{...})$$
$$t^{ ...}_{\beta ...}=\frac{1}{2}(t^{ ...}_{b...}-i J^c_bt^{ ...}_{c...}), \:\:t^{ ...}_{\overline{\beta} ...}=\frac{1}{2}(t^{ ...}_{b...}+i J^c_bt^{ ...}_{c...}).$$
The first question is whether one knows a reference for precise elaboration on that notation since in mentioned place I only found hardly this formulas.
The second and most important question is how this correspond to the usual notation in which $Greek$ and $\overline{Greek}$ letters correspond to holomorphic and anti-holomorphic decomposition.
Let me elaborate on that second question. For a fixed tensor t of type $(p,q)$ we have its decomposition into $2^{p+q}$ tensors being sections of $${T^{1,0}}^{P'}M \otimes {T^{1,0}}^{P''} M \otimes {{T^{1,0}}^*}^{Q'}M \otimes {{T^{1,0}}^*}^{Q''}M$$ where $P' \cup P'' = [p]$, $Q' \cup Q'' = [q]$ and $P',P'',Q',Q''$ actually indicate order in which $i$ and $-i$ eigenspaces are tensored.
It seems to me (I would as well appreciate an easy reason for that) that for a fixed combination of bars and no-bars, $t^{\alpha \overline{\beta}...}_{\gamma \delta...}$ are coordinates for the tensor being a member of decomposition of $t$ corresponding to this "type" but still in coordinates $(U,x_1,...,x_{2m})$.
Suppose now in addition that we have started with a holomorphic coordinates i.e. $x_j+ix_{m+j}$ are holomorphic coordinates. How are then this $t^{\alpha ...}_{...}$ related to the components of $t$ in holomorphic coordinate i.e. with respect to $\partial_{z_j}, \partial_{\overline{z_j}}, dz_j, d\overline{z_j}$.
For example for Hermitina metric $g$ is $g_{\alpha \overline{\beta}}$ in usual sense, i.e. $g=g_{\alpha \overline{\beta}}(dz_\alpha \otimes d\overline{z_\beta} + d\overline{z_\beta} \otimes dz_\alpha)$, the same as in the one defined in the beginning?
Note that in formulas from the begging, Greek letters can be choose to be bigger than $m$ so it's probably not that this components are the same on the other hand latter in the book in case of holomorphic coordinates Joyce is using them as they were the same.