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In the book 'Principles of Algebraic Geometry' P141 and the book 'Differential Forms in Algebraic Topology' P72-73, they defined the Chern(Euler) classes of line bundles using the patching data $\{U_\alpha\}$ and $g_{\alpha\beta}$.

However, the 2-forms they give above differed by a negative sign. In [BT], the form is given by $\epsilon_{\alpha\beta\gamma}=-\frac{i}{2\pi}(\log g_{\alpha\beta}+\log g_{\beta\gamma}-\log g_{\alpha\gamma})$, they called it Euler class but we all know that in line bundle case Euler class equals to Chern class. But in [GH], the Chern class of a line bundle is given by $z_{\alpha\beta\gamma}=\frac{i}{2\pi}(\log g_{\alpha\beta}+\log g_{\beta\gamma}-\log g_{\alpha\gamma})$. So all I see is that their definitions seems differed by a negative sign. Yet they all claim when you consider a Riemann surface, the integration of Euler class of tangent bundle over the Riemann surface will give the Euler characteristic.

I don't know what's wrong about that. Any comment would be grateful.

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