Holomorphic line bundle

A holomorphic line bundle is a complex line bundle with holomorphic local trivializations and holomorphic nonvanishing transition functions.

A global holomorphic section of $\mathcal{O}(1)$ on ${\mathbb{CP}}^1$ is a linear homogeneous polynomial in the two projective coordinates. On one patch its coefficient is an entire function $p(\lambda )$; regularity on the other patch says $p(\lambda )/\lambda$ is regular at infinity. Hence $p$ has at most linear growth. Cauchy estimates make its second and higher derivatives zero, so $p=a+b\lambda$. The same argument entrywise applies to matrix-valued sections and makes the logarithmic derivative in the Penrose-Ward correspondence linear in the primed two-component spinor.

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.

A nowhere-zero smooth section $s$ of a holomorphic line bundle writes its Dolbeault operator as ${\bar{\partial }}_{L}s=a\otimes s$. Its integrability gives $\bar{\partial }a=0$. If $H^{0,1}(X)=0$, then $a=\bar{\partial }f$ globally. The section $e^{-f}s$ is nowhere zero and holomorphic. This argument requires no compactness and explicitly constructs the holomorphic trivialization.

In a holomorphic local frame, a meromorphic section has a meromorphic scalar coefficient; the coefficients transform by the bundle's transition functions of a vector bundle. A nonzero meromorphic section has a divisor on a complex manifold recording its zero and pole orders.

For a holomorphic line bundle on a complex curve, a meromorphic section modulo sections holomorphic through a point is its principal part. In a local coordinate and holomorphic local frame it is represented by finitely many negative powers. A vanishing $H^1(X,\mathcal{O}(E))$ permits prescribed principal parts, by a cutoff followed by solving a global Dolbeault operator equation.

For the structure sheaf of a complex manifold, local holomorphic logarithms give the short exact sequence of sheavesThe connecting map $H^1(M,{\mathcal{O}}_M^{\ast })\to H^2(M,\mathbb{Z})$ sends a holomorphic line bundle to its First Chern class. In particular, it is onto when $H^2(M,{\mathcal{O}}_M)=0$.

A Dolbeault partial connection obeys ${\bar{\partial }}_E(fs)=\bar{\partial }f\otimes s+f{\bar{\partial }}_{E}s$. Its square is a tensorial $(0,2)$-form with values in $\mathrm{End}E$; it vanishes for the canonical partial connection of a holomorphic bundle.