 Curvature Dolbeault class

Chern curvature defines $\alpha (E)=[\Theta ]\in H^1(M,{\Omega }_M^1\otimes \mathrm{End}E)$. If two metrics give connections differing by $a$, then ${\Theta }_1-{\Theta }_0={\bar{\partial }}_{\mathrm{End}E}a$. Thus this vector-bundle curvature class depends on the holomorphic bundle rather than the chosen metric.