# Do we have a $4$-term decomposition of $\bar{\partial}_M$ for a holomorphic fiber bundle $M\to B$?

First let us consider a Riemannian fiber bundle, i.e a fiber bundle $\pi: M\to B$ of oriented Riemannian manifolds. We denote by $T(M/B)$ the bundle of vertical tangent vectors and assume that the bundle $\pi: M\to B$ possesses the following additional structures:

1. a connection, that is, a choice of a splitting $TM=T_HM\oplus T(M/B)$ so that the subbundle $T_HM$ is isomorphic to the vector bundle $\pi^*TB$;
2. a connection $\nabla^{M/B}$ on $T(M/B)$.

Let $P: TM\to T(M/B)$ be the projection operator with kernel the chosen horizontal tangent space $T_HM$.

Let $S$ be the second fundamental form as a section of the bundle $$End(T(M/B))\otimes T_H^*M\cong T^*(M/B)\otimes T(M/B)\otimes T^*_HM$$ defined by $$(S(X,\theta),Z)=\langle\nabla^{M/B}_ZX-P[Z,X],\theta\rangle$$ for $Z\in \Gamma(M,T_HM)$, $X\in \Gamma(M,T(M/B))$ and $\theta\in \Gamma(M,T^*(M/B))$.

Let the tensor $\Omega$ be the section of the bundle $Hom(\wedge^2T_HM,T(M/B))$ over $M$ defined by the formula $$\Omega(X,Y)=-P[X,Y]$$ for $X$ and $Y$ in $\Gamma(M,T_HM)$.

We could check that $S$ and $\Omega$ are tensors.

The de Rham differential $d_M$ may be expressed in thers of $\nabla^{M/B}$, $S$, $\Omega$, and the vertical exterior differential $d_{M/B}$ of the fiber bundle. First we extend $d_{M/B}$ to an operator on $\Gamma(M,\wedge T^*_HM\otimes \wedge T^*(M/B)$ by the formula $$d_{M/B}(\pi^*\nu\otimes \beta)=(-1)^{|\nu|}\pi^*\nu\otimes d_{M/B}\beta$$ for $\nu\in \mathscr{A}(B)$ and $\beta\in \Gamma(M,\wedge T^*(M/B))$.

Let $e_i$ be a local frame of $T(M/B)$ and $f_{\alpha}$ be a local frame of $TB$, with dual frames $e^i$ and $f^{\alpha}$, respectively. We define an operator $\delta_B$ by $$\delta_B(\pi^*\nu\otimes \beta)=\pi^*(d_B\nu)\otimes \beta+(-1)^{|\nu|}\pi^*\nu\otimes \sum f^{\alpha}\wedge \nabla^{M/B}_{f_{\alpha}}\beta$$

Then $d_M$ could be decomposed as $$\tag{*}d_M=d_{M/B}+\delta_B-\sum\langle S,e^i\rangle \iota(e_i)+\sum\langle \Omega,e^i \rangle\iota(e_i).$$

See Heat Kernels and Dirac Operators Section 10.1, in particular Proposition 10.1.

Now we consider a holomorphic fiber bundle of complex Riemannian manifolds $\pi^*: M\to B$. Let $J$ and $J^{\prime}$ be the complex structure on $TM$ and $TB$ respectively. $J$ maps $T(M/B)$ into itself. We also assume that $J$ maps $T^HM$ into itself. However we do not require $T^{H,(1,0)}M$ to be a holomorphic subbundle of $T^{(1,0)}M$. Moreover we assume the fiber bundle is Kahler in the sense that there exists a smooth $2$-form $\omega$ on $M$ of complex type $(1,1)$ with the following properties

1. $\omega$ is closed;
2. $T^HM$ and $T(M/B)$ are orthogonal with respect to $\omega$;
3. If $X$ and $Y\in T(M/B)$, then $\omega(X,Y)=\langle X,JY\rangle$ where the right hand side is given by the Riemannian metric. See Analytic Torsion and Holomorphic Determinant Bundles II Section 1.(c).

My question is: in the complex case, do we have a $4$-term decomposition of $\bar{\partial}_M$ which is the analogue of the decomposition of $d_M$ in (*)?