Let $L$ be a holomorphic line bundle on a compact complex manifold $X$ with $\dim_{\mathbb C} X = n$. The first Chern class $c_1 (L)$ can be directly related to the curvature of a connection on $L$ by $$ c_1 (L) = \left[ \frac{i}{2 \pi} \Theta \right] \in \mathrm H^2_{\mathrm{dR}} (X) $$ where $\Theta$ is the curvature form of the connection.
Can the first Chern class also be interpreted as some suitable notion of curvature of the total space $M = \mathrm{Tot} (L)$ of the line bundle, which would be a (real) manifold of dimension $2n + 2$?