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Let $X$ be a smooth projective complex curve. Consider the diagonal $\Delta$ in $X \times X$, and $\mathcal{O}(\Delta)$ the associated line bundle. If $j$ is the inclusion of $\Delta$ in $X \times X$ and $TX$ is the tangent bundle to $X$, there is a canonical map from $\mathcal{O}(\Delta)$ to $j_*(TX)$. Is this map surjective? If not, what is the image?

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  • $\begingroup$ It is surjective. For any line bundle $L$ on $X\times X$, the natural map $L\rightarrow j_*j^*L$ is surjective (just look locally). For $L=\mathcal{O}(\Delta )$, this is the map you consider. $\endgroup$
    – abx
    Nov 20, 2019 at 18:10
  • $\begingroup$ Thank you for the answer. And what about the map from $p_*(\mathcal{O}(\Delta))$ to TX, where p is one projection from XxX to X? $\endgroup$
    – user95246
    Nov 21, 2019 at 12:58
  • $\begingroup$ This map is zero. $\endgroup$
    – user95246
    Nov 21, 2019 at 15:10

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