= Two-section criterion for holomorphic triviality
{title2=$H^0(X,L)\ne0,\ H^0(X,L^*)\ne0\Longleftrightarrow L\cong\mathcal O_X$}
On a compact connected <complex manifold>, pair nonzero <holomorphic sections> of a <holomorphic line bundle> and its <holomorphic dual vector bundle>. Their product is a global <holomorphic function>, hence constant by the <maximum modulus principle>. The <identity theorem> makes their nonvanishing loci dense, so this constant is nonzero. Both sections are consequently nowhere zero and trivialize their bundles. Conversely, the constant section of a trivial bundle and its dual provide the two sections.
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