d c operator ($d^c$)

With the convention $d^c=i(\bar{\partial }-\partial )$ and with $J$ acting on forms by applying the complex structure to every argument,Indeed, $J$ acts on a $(p,q)$-form by $i^{p-q}$; conjugating $d=\partial +\bar{\partial }$ therefore multiplies its two type components by $-i$ and $i$, respectively.