exact sequences give us boundary isomorphisms $$\frac{H^0(\mathcal{Z}^2)}{dH^0(\mathcal{A}^1)}\xrightarrow{\delta_1}H^1(\mathcal{Z}^1_d),\qquad
H^1(\mathcal{Z}^1_d) \xrightarrow{\delta_2} H^2({\mathbb{C}}).$$ As explained in *loc. cit.* the multiple of the Chern class $-(2\pi i)c_1({\mathcal{L}})$ can be calculated as $\delta_2\delta_1(\Theta)$, where $\Theta$ is the curvature form of a connection on ${\mathcal{L}}$. It also follows from *loc. cit.* that $\delta_1(\Theta)$ is represented by the cocycle $-d\log g_{ij}$.

\[propc1L=0\] Let $X$ be a smooth projective variety over ${\mathbb{C}}$. If ${\mathcal{L}}$ is a line bundle such that $H^1(X,{\mathcal{L}})=0$ and ${\mathcal{L}}\xrightarrow{\omega}\Omega^1_X$ is a foliation without persistent singularities then $c_1({\mathcal{L}})=0$, where $c_1({\mathcal{L}})$ is the Chern class of the line bundle viewed in $H^2(X,{\mathbb{C}})$.

Let $(U_i,\phi_i)$ be a trivialization of ${\mathcal{L}}$ with gluing data $g_{ij}\in {\mathcal{O}}^*_X(U_{ij})$. On each $U_i$ we have a local generator of ${\mathcal{L}}(U_i)$, namely $\phi_i^{-1}(1)$, we denote by $\omega_i$ the image under $\omega$ of this generator. The fact that the foliation defined by $\omega$ has no persistent singularities means that on each $U_i$ there is a local section $\eta_i$ of $\Omega^1_X/{\mathcal{L}}(U_i)$ such that $d\omega_i=\omega_i\wedge\eta_i$. On $U_{ij}$ the restriction of the local $1$-form $\omega_i$ satisfies $$\omega_i=g_{ij}\omega_j.$$ So computing the de Rham differential of this forms on $U_{ij}$ gives us, $$\begin{aligned}
\omega_i\wedge\eta_i=d\omega_i=d(g_{ij}\omega_j)=\\
=g_{ij}d\omega_j+dg_{ij}\wedge \omega_j=\\
=g_{ij} \omega_j\wedge\eta_j + dg_{ij}\wedge \omega_j=\\
=g_{ij}\omega_j\wedge\left(\eta_j-\frac{dg_{ij}}{g_{ij}}\right).\end{aligned}$$ Subtracting both sides of the equality we 