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Let $K$ be the canonical line bundle of a compact Riemann surface $M$ of genus $g$. Consider the pull back of $K$ on $M \times M$ via projection on the first factor. What is the dimension of the space of global holomorphic sections of this pull back line bundle on $M \times M$?

I can calculate its Euler characteristic by Riemann–Roch (Noether’s formula), but how does one calculate $H^0$?

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    $\begingroup$ By the Kunneth formula: $H^0(M\times M,K\boxtimes \mathscr{O}_M)=H^0(M,K)\otimes H^0(M,\mathscr{O}_M)$ $\cong H^0(M,K)$, hence the dimension is $g$. $\endgroup$
    – abx
    Commented Dec 5, 2020 at 19:42
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    $\begingroup$ See mathoverflow.net/questions/120033/… $\endgroup$
    – Margaret Friedland
    Commented Dec 5, 2020 at 19:52

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