Two-section criterion for holomorphic triviality ($H^0(X,L)\ne 0,H^0(X,L^{\ast })\ne 0⟺L\cong {\mathcal{O}}_X$)

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.