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A holomorphic line bundle $L\to X$ is a rank-one holomorphic vector bundle: it has local trivializations $L{|}_{U_i}\cong U_i\times \mathbb{C}$ whose fiberwise linear transition functions of a vector bundle are nowhere-zero holomorphic functions. Equivalently, choose local holomorphic frames $e_i$ and write $e_j=g_{ij}{e}_i$ on overlaps, with $g_{ij}\in {\mathcal{O}}^{\ast }(U_i\cap U_j)$ and the cocycle relation on triple overlaps. A holomorphic section over $U$ has coefficients $s_i$ holomorphic on $U\cap U_i$ in $s=s_i{e}_i$; consistency means $s_i=g_{ij}{s}_j$.

The dual bundle has fiber $L_x^{\ast }={\mathrm{Hom}}_{\mathbb{C}}(L_x,\mathbb{C})$. Its dual frames $e_i^{\ast }$ are characterized by $e_i^{\ast }(e_i)=1$, and the frame relation becomes $e_j^{\ast }=g_{ij}^{-1}{e}_i^{\ast }$. Reciprocals of nowhere-zero holomorphic functions are holomorphic, and their cocycle relations persist. This defines the holomorphic dual vector bundle structure on $L^{\ast }$, with holomorphic evaluation pairing $L^{\ast }\otimes L\to {\mathcal{O}}_X$.

Assume now that $X$ is compact and connected, and let $s$ and $t$ be nonzero global holomorphic sections of $L$ and $L^{\ast }$, respectively. The function $t(s)$ is a global holomorphic function. The maximum modulus principle, applied in local complex coordinate lines near a maximum and then propagated on the connected manifold, makes it constant. Moreover it is not identically zero: the zero locus of a nonzero holomorphic section has empty interior by the identity theorem, so the two nonvanishing loci are dense open sets and intersect. Thus $t(s)=c\ne 0$ everywhere.

In particular, $s$ never vanishes. The map $(x,a)\mapsto as(x)$ is a holomorphic bundle isomorphism $X\times \mathbb{C}\to L$; in a local frame its inverse divides by the nowhere-zero holomorphic coefficient of $s$. Conversely, a holomorphic trivialization supplies the constant unit sections of both $L$ and $L^{\ast }$. We have proved the two-section criterion for holomorphic triviality:

Write points of Complex projective space as lines $[Z]\subset {\mathbb{C}}^{n+1}$. The complex tautological line bundle iswith fiber that very line. Its dual is the hyperplane line bundle $\mathcal{O}(1)$. On the affine chart $U_i=\{Z_i\ne 0\}$, the vector $e_i([Z])=Z/Z_i$ is a nowhere-zero holomorphic frame of $\mathcal{O}(-1)$. On $U_i\cap U_j$,Both ratios are nowhere-zero holomorphic functions of the affine coordinates. They exhibit the holomorphic transition functions of a vector bundle for the two bundles explicitly.

Choose a nonzero linear functional $\ell :{\mathbb{C}}^{n+1}\to \mathbb{C}$ whose zero hyperplane is $H$. At $[Z]$, its restriction to $\mathbb{C}Z$ is an element of that line's dual and therefore defines a holomorphic section $s_{\ell }$ of $\mathcal{O}(1)$. In the dual frame above its coefficient isThis expression is holomorphic even along $H$, and its zero locus is exactly $H$. Consequently $s_{\ell }{|}_{{\mathbb{P}}^n\setminus H}$ is the requested nowhere-zero section; its global holomorphic extension vanishes precisely on $H$. The extension need not, and does not, stay nowhere zero on the hyperplane.

Use the convention that a Hermitian metric $h$ is complex-linear in its first argument and conjugate-linear in its second. It is a smoothly varying positive-definite Hermitian form on each fibre. A connection on a vector bundle is a complex-linear operator $D:\Gamma (E)\to {\Omega }^1(X,E)$ satisfying $D(as)=da\otimes s+aDs$ for smooth complex functions $a$. Metric compatibility means, for every real vector field $V$,This is the metric-compatible connection condition with the sesquilinear convention fixed.

Apply the smooth Gram-Schmidt process to a local frame to obtain an $h$-orthonormal smooth frame $s_1,\dots ,s_r$. Positivity guarantees that all normalization denominators are nonzero and depend smoothly on the base point. Write $D{s}_j={\sum }_i{A}_{ij}{s}_i$. Differentiating $h(s_i,s_j)={\delta }_{ij}$ and using compatibility givesfor every real $V$. Thus $\boxed{A(V{)}^{†}=-A(V)}$: the connection matrix is skew-Hermitian. This smooth unitary frame for a Hermitian connection is generally not holomorphic; the requested local-frame assertion requires only a smooth frame.

The holomorphic dual vector bundle $E^{\ast }$ is obtained by dualizing fibres and using transition matrices $g_{ij}^{-T}$ when $E$ has transitions $g_{ij}$. Their entries are holomorphic because matrix inversion is holomorphic on $G{L}_r(\mathbb{C})$. Their cocycle property follows from preservation of the fibrewise evaluation pairing, defining the natural holomorphic bundle with fibre $(E_x{)}^{\ast }$.

The conjugate vector bundle $\overline{E}$ has the same underlying real fibres but opposite scalar action: $\lambda \cdot \overline{v}=\overline{\overline{\lambda }v}$. Its smooth transition matrices are $\overline{g_{ij}}$; they need not be holomorphic on $X$. The conjugate bundle is used here as a smooth complex bundle, whereas $E^{\ast }$ is holomorphic.

Define the smooth dual tensor $H$ byIt is complex-linear in each tensor factor, because conjugating the second bundle converts the conjugate-linearity of $h$ into linearity. Thus $H\in \Gamma ((E\otimes \overline{E}{)}^{\ast })$.

The conjugate connection is defined on real vector fields by ${\overline{D}}_V(\overline{s})=\overline{D_{V}s}$ and extended complex-linearly on the conjugate bundle. The tensor product connection isIts dual connection $D_0$ is uniquely characterized byThe Leibniz rule makes this a genuine connection on $(E\otimes \overline{E}{)}^{\ast }$. Evaluate it on the decomposable tensor $u=s\otimes \overline{t}$:Decomposable tensors span every fibre. Hence this tensor-valued one-form vanishes precisely when the compatibility identity holds for every $V,s,t$:This is metric compatibility as parallelism of a Hermitian tensor.

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.