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Let $X$ be a connected complex projective manifold, $\chi:\pi_1(X)\to S^1$ be a character of the fundamental group of $X$. Then $\chi$ induces a local system $\mathcal{L}_{\chi}$ of rank $1$ on $X$ and $L=\mathcal{L}_{\chi}\otimes_{\mathbb{C}}O_X$ is a line bundle on $X$. Since $L$ is flat, the real Chern class $c_1^{\mathbb{R}}(L)=0$ in $H^2(X,\mathbb{R})$. What is the integral Chern class $c_1^{\mathbb{Z}}(L)$ in $H^2(X,\mathbb{Z})$? It must be an torsion element, but is there an explicit formula?

Doug
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    The local system can be viewed as a principal S^1-bundle, that is, as a cohomology class A in H^1(X, S^1). The first Chern class of the associated line bundle is just the image of A under the boundary map H^1(X, S^1) —> H^2(X, Z) – Cranium Clamp Aug 29 '23 at 11:31
  • Dear @CraniumClamp, Thank you for your answer! I think you are right. But a sign may be there due to various conventions; see Prop. 4.4.12 of Complex Geometry: An Introduction by D. Huybrechts. – Doug Aug 29 '23 at 14:08

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