The brief setting of the problem is the following. Suppose we are given the Poincare metric on the open modular curve, and the metric is singular at cusps. The Chern form of the tangent bundle defined by the metric, if we view it as a current, has finite integrals on the compactified modular curve. According to the paper I read, with this metric, the tangent bundle has a canonical extension to the compactified curve. Does that mean I need to find a line bundle with the same Chern class as the current previously defined on the compact curve?
Let $\Gamma = \Gamma(N)$, $X_{\Gamma}^0$ be the open modular curve defined by $\mathcal{H}/\Gamma$ where $\mathcal{H}$ is the upper half plane, $X_{\Gamma}$ be the compactified modular curve at cusps. We have the Poincar'e metric $$\frac{1}{y^2}d\tau \wedge d\bar{\tau}$$ on the upper half plane. Using the covering map $\tau\mapsto e^{i\tau}$ (or $\tau\mapsto e^{\frac{2\pi i\tau}{N}}$, no matter which is used, the form of metric written done is the same), the cusp at infinity has a nbd which is biholomorphic to a unit disk. Then on the punctured unit disk, and the tangent bundle of $X_\Gamma$ has a singular Hermitian metric descending from the Poincar'e metric. Let $w=e^{iz}$ be the coordinate at infinity, the metric is $$\frac{1}{|w|^2(\log |w|)^2} dw \wedge d\bar{w}.$$ Computing the Chern form of tangent bundle, near infinity: \begin{align*} \frac{\sqrt{-1}}{2\pi} \bar{\partial} \partial \log (\frac{1}{|w|^2 (\log |w|)^2}) & = \frac{\sqrt{-1}}{2\pi} \partial \bar{\partial} \log(|w|^2) + \frac{\sqrt{-1}}{2\pi} \partial \bar{\partial} \log (\log |w|)^2 \end{align*} The first term on the RHS, viewed as a current, by thr Poincar'e-Lelong equation, it is equal to the evaluation current at infinity with multiplicity $+1$, and the second has finite integration for forms supported near infinity. Then define $$\omega := c_1(T_{X_{\Gamma}^0}),$$ viewed as a current, is well-defined on the compactified modular curve $X_{\Gamma}$. I wonder how to find a line bundle, whose first Chern class has the same cohomology class as $\omega$? The paper I read says that the extension should be $T_{X_\Gamma} \otimes [-q_1 -\cdots -q_r]$, where $q_i$ are cusps. But in the above equation, the $(\partial \bar{\partial} \log\log)$ term makes me confused to tell why the extension should be $T_{X_\Gamma} \otimes [-q_1 -\cdots -q_r]$. Also, my calculation shows that the cusp at infinity should have multiplicity $+1$ in the extension, not $-1$.
Or is there something wrong about my understanding on extending a line bundle using a singular Hermitian metric on an open Riemann surface to the compactified Riemann surface? My knowledge in the theory of current is very limited, and I think I should also read something further in this direction. I want to find a detailed reference focusing on application of currents on complex manifolds. I know there is a book by Pierre Lelong, but it is in French. Any other reference is welcomed!