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Suppose $\mathbb T:=V/\Gamma$ is a complex $n$-torus (i.e., $V$ is an $n$-dimensional $\mathbb C$-vector space and $\Gamma$ is a rank $2n$ lattice in $V$). Fix a holomorphic line bundle $L\in\text{Pic}(\mathbb T)$. Now, consider the holomorphic map $f:\mathbb T\to \text{Pic}^0(\mathbb T)$ given by $\mu\mapsto L^{-1}\otimes\tau_\mu^*L$. Here, $\tau_\mu$ is the map $z\mapsto z+\mu$ on $\mathbb T$. Being a map of complex tori, with $0\mapsto 0$, $f$ is completely, determined once the induced map $f_*:H_1(\mathbb T,\mathbb Z)\to H_1(\text{Pic}^0(\mathbb T),\mathbb Z)$ is determined.

On pages 316-317 of their book "Principles of Algebraic Geometry", Griffiths-Harris prove that for an ample $L$, the map $f_*:\Gamma\to\text{Hom}(\Gamma,\mathbb Z)$ is given by the (non-degenerate, skew symmetric) pairing $c_1(L):\Gamma\times\Gamma\to\mathbb Z$. Their proof heavily relies on ampleness to choose a special coordinate system in which the computations are performed.

Is the result true for non-ample line bundles as well, and is there a less computational proof of this fact (even for the case of ample line bundles)?

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  • $\begingroup$ Since the space of holomorphic line bundles with a given Chern class is connected, it is clear a priori that the map $f_*:\Gamma\to\text{Hom}(\Gamma,\mathbb Z)$ depends only on $c_1(L)$. $\endgroup$ – Mohan Swaminathan Apr 15 '18 at 19:49
  • $\begingroup$ Look at Mumford's Abelian varieties, §8. $\endgroup$ – abx Apr 15 '18 at 20:04

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