R(X, Y, JZ, W) + R(X, Y, Z, JW).$$ Since ${\hat{{\operatorname{P}}}}= -{\operatorname{F}}^2$ and ${\tilde{{\operatorname{P}}}}= \operatorname{Id}- {\hat{{\operatorname{P}}}}$ by Lemma \[lem:projprop\], we also have ${\operatorname{R}}\circ {\hat{{\operatorname{P}}}}= 0$ and ${\operatorname{R}}\circ {\tilde{{\operatorname{P}}}}= {\operatorname{R}}$. Then $$0 = \nabla ({\operatorname{R}}\circ {\operatorname{F}}) = \nabla {\operatorname{R}}\circ {\operatorname{F}}+ {\operatorname{R}}\circ \nabla {\operatorname{F}}= \nabla {\operatorname{R}}\circ {\operatorname{F}},$$ and the other identities in follow similarly.

The identities in Lemma \[lem:init\] reflect that the image of the curvature operator is contained in the parallel subalgebra $\mathfrak{hol}(g(0))$ of ${\wedge^2 T^*M}$ defined by the holonomy representation of $g(0)$. Since $g(0)$ is Kähler, this means that, at any $p\in M$, $$\operatorname{im}{\operatorname{R}}(p, 0) \subset\mathfrak{hol}_p(g(0)) \subset \mathcal{H}_p(0) \cong \mathfrak{u}(n/2),$$ and so $$\mathcal{K}_p(0) = \mathcal{H}_p^{\perp}(0) \subset \operatorname{ker} {\operatorname{R}}(p, 0),$$ since ${\operatorname{R}}(p, 0)$ is self-adjoint.

A closed system of mixed differential inequalities
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Now define the sections $${\operatorname{A}}= \nabla {\hat{{\operatorname{P}}}}\in T^*M\otimes{\operatorname{End}}({\wedge^2 T^*M}), \quad {\operatorname{B}}= \nabla\nabla {\hat{{\operatorname{P}}}}\in T^*M\otimes T^*M\otimes {\operatorname{End}}({\wedge^2 T^*M}).$$ It follows from the general computations in Section 4 of [@KotschwarRFHolonomy] that ${\hat{{\operatorname{R}}}}$ and ${\hat{{\operatorname{S}}}}$, together with ${\operatorname{A}}$ and ${\operatorname{B}}$, satisfy a closed system of differential inequalities.

\[prop:sysev\] The sections ${\operatorname{A}}$, ${\operatorname{B}}$, ${\hat{{\operatorname{R}}}}$, ${\hat{{\operatorname{S}}}}$ satisfy $$\begin{aligned}
\label{eq:aev}
      \left|D_{\tau} {\operatorname{A}}\right| &\leq C|{\operatorname{R}}||{\operatorname{A}}| + C|{\hat{{\operatorname{S}}}}|\\  
\label{eq:bev}
     \left|D_{\tau}{\operatorname{B}}\right| &\leq C|\nabla {\operatorname{R}}| |{\operatorname{A}}|+ C|{\operatorname{R}}|\left(|{\operatorname{B}}| + | \nabla {\hat{{\operatorname{S}}}}|\right) \\
 \label{eq:rhev}
 \left|\left(D_{\tau} + \Delta\right){\hat{{\operatorname{R}}}}\right| &\leq C|\nabla 