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Aug 23, 2020 at 16:07 comment added abx The standard definition of the Jacobian is $\Omega (X)^*/H_1(X,\mathbb{Z})$, where $H_1(X,\mathbb{Z})$ embeds into $\Omega (X)^*$ by $\gamma \mapsto \int_{\gamma }$. This is of course independent of any choice, and choosing bases leads to the formulation you mention. For this kind of basic question please consider using MSE.
Aug 23, 2020 at 15:11 history edited user267839 CC BY-SA 4.0
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Aug 23, 2020 at 4:26 history asked user267839 CC BY-SA 4.0