Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 126 / 3 / ii / Solution

For a line bundle $\mathcal{L}$ on $X$, define the homomorphism associated to a line bundle on an abelian varietyThe translation identity from part (i) givesso ${\varphi }_{\mathcal{L}}$ is a homomorphism.

Suppose $\mathcal{M}\simeq T_y^{\ast }\mathcal{L}\otimes {\mathcal{L}}^{\vee }$ lies in its image. Thenwhich is trivial by the same translation identity. Hence ${\varphi }_{\mathcal{M}}=0$; the image of every ${\varphi }_{\mathcal{L}}$ lies in the Identity component of the Picard group ${\mathrm{Pic}}^{0}X$.