Ricci flow). Instead we will specialize to the Kähler setting the approach in [@KotschwarRFHolonomy], which encodes the general problem of preservation of reduced holonomy in terms of the uniqueness of solutions to an associated system of mixed differential inequalities.

A family of almost-complex structures on $M$.
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The first step in our reduction of the problem is to extend $J_0$ to a family of almost complex structures $J = J(\tau)\in \operatorname{End}(TM)$ for $\tau\in [0, T]$ relative to which $g(\tau)$ remains Hermitian. (In our application, we will eventually we will wish to argue that $J(\tau) = J_0$.) We do so by solving the ordinary differential initial value problem $$\label{eq:jev}
  \left\{\begin{array}{rl}
          {{\frac{\partial }{\partial\tau}}}J^a_b =  R_b^c J^a_c - R_c^aJ_b^c & \mbox{on}\ M\times (0, T]\\
          J(x, 0) = J_0(x) & \mbox{on}\ M
         \end{array}\right.$$ in each fiber of $\operatorname{End}(TM)$.

The evolution equation in may be understood in terms of the operator $D_{\tau}$ which acts on families $V = V(\tau)$ of $(k, l)$-tensors by $$\begin{aligned}
 D_{\tau}V_{b_1b_2 \ldots b_k}^{a_1a_2\ldots a_l} &= {{\frac{\partial }{\partial\tau}}}V_{b_1b_2 \ldots b_k}^{a_1a_2\ldots a_l} 
 - R_{b_1}^{c}V_{cb_2\ldots b_k}^{a_1a_2\ldots a_l}  - R_{b_2}^{c}V_{b_1c\ldots b_k}^{a_1a_2\ldots a_l}
 - \cdots - R_{b_k}^{c}V_{b_1b_2\ldots c}^{a_1a_2\ldots a_l}\\
 &\phantom{={{\frac{\partial }{\partial\tau}}}V_{b_1b_2 \ldots b_k}^{a_1a_2\ldots a_l} }+ R_{c}^{a_1}V_{b_1b_2\ldots b_k}^{ca_2\ldots a_l}
 + R_{c}^{a_2}V_{b_1b_2\ldots b_k}^{a_1c\ldots a_l} + \cdots + R_{c}^{a_l}V_{b_1b_2\ldots 