In Huybrecht's book on Fourier-Mukai transforms the following argument is used in a proof
Since $(X,\mathcal L)$ is a principally polarized abelian variety one has a unique global section $s : \mathcal O_X \to \mathcal L$.
The only things that seem to be used here is that a $\mathcal L$ is a principal polarization, so an ample line bundle such that $\chi(\mathcal L) = 1$.
This argument seems very brief to me. Can anyone explain to me this argument in more detail?
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This seems to follow from the fact (Mumford, Abelian Varieties) that $\chi(\mathcal L) = \text{dim}_k\text{H}^0(X,\mathcal L)$ (of which I have no proof).
Samir and Ruben have mentioned in the comments that this follows from Kodaira vanishing. I'll type out an answer (I myself found this instructive).
Kodaira vanishing in its algebraic form asserts that for $X$ a smooth projective $k$-scheme of dimension $d$ over a field of characteristic zero, and $\mathcal{L}$ ample, we have $$ H^q(X, \mathcal{L} \otimes \Omega^p_{X/k} ) = 0 \qquad \text{ for } \qquad p+q > d. $$ Since $X$ is an abelian variety, its cotangent sheaf is trivial, i.e. $\Omega_{X/k} \cong \mathcal{O}_X$. But then in particular we have $H^q(X, \mathcal{L} \otimes \Omega^d_{X/k} ) \cong H^q(X, \mathcal{L} ) = 0 $ for $d+q>d$, i.e. for $q>0$. It then follows by definition of the Euler characteristic of a sheaf, $$ \chi(\mathcal{L}) = (-1)^i \sum_i \dim_k H^i(X, \mathcal{L}) $$ that $\chi(\mathcal L) = \text{dim}_k\text{H}^0(X,\mathcal L)$.