Dolbeault partial connection (${\bar{\partial }}_E$)

A Dolbeault partial connection obeys ${\bar{\partial }}_E(fs)=\bar{\partial }f\otimes s+f{\bar{\partial }}_{E}s$. Its square is a tensorial $(0,2)$-form with values in $\mathrm{End}E$; it vanishes for the canonical partial connection of a holomorphic bundle.