My question concerns a particular preliminary remark made at the bottom of page 33 of https://arxiv.org/abs/1803.06697.
Let $f : X \to B = B_1(0) \subset \mathbb{C}^m$ be a surjective, proper, holomorphic submersion with Calabi-Yau fibres $Y$ and total space a compact Kähler manifold $X$ equipped with a Ricci-flat metric $\omega_X$. By Ehresmann's theorem, $f$ is a $C^{\infty}$ fibre bundle, and up to shrinking $B$, we may choose a smooth trivialisation $\Phi : B \times Y \to X$, where $Y = f^{-1}(0)$ is viewed as a smooth real $2n$-dimensional manifold. We then equip $B \times Y$ with the complex structure $J^{\sharp}$ induced by the one on $X$ via $\Phi$, so that $\Phi$ becomes a biholomorphism and the projection onto $\mathbb{C}^m$ becomes a $J^{\sharp}$-holomorphic submersion. Let $J_{Y,z}$ denote the restriction of $J^{\sharp}$ to the $J^{\sharp}$-holomorphic fibre $\{ z \} \times Y$ and let $J_z = J_{\mathbb{C}^m} + J_{Y,z}$.
Now fix a smooth complex coordinate chart $(y^1, ..., y^n)$ on $Y$ such that $(z^1, ..., z^m, y^1, ..., y^n)$ is a smooth complex coordinate chart of $B \times Y$. I am struggling to understand the following:
Schematically, and ignoring the distinction between these complex coordinates and their complex conjugates, we may then write $$(J^{\sharp} - J_{z_0})\vert_{(z,y)} = A(z_0, z,y) \ast dz \otimes \partial_y + B(z_0, z,y) \ast dy \otimes \partial_y,$$ where $\ast$ denotes tensor contraction, $A,B$ are smooth matrix valued functions with $B(z_0, z_0, y)=0$. Moreover, there are no $dz \otimes \partial_z$ or $dy \otimes \partial_z$ terms because the projection onto $\mathbb{C}^m$ is holomorphic with respect to both $(J^{\sharp}, J_{\mathbb{C}^m})$ and $(J_{z_0}, J_{\mathbb{C}^m})$.
Q1: Can someone help me write out the entire expression, without ignoring the distinction between the complex coordinates and their conjugates?
Q2: Why does the projection onto the $\mathbb{C}^m$ factor being holomorphic omit the above terms.