Holomorphic section

A section of a holomorphic vector bundle is holomorphic when its coordinate functions in every holomorphic local trivialization are holomorphic.

It is a section of a fiber bundle satisfying holomorphicity.

A section of $\mathcal{O}(1)\oplus \mathcal{O}(1)$ on the complex projective line is a pair of homogeneous linear forms ${\ell }_1=a{Z}_0+b{Z}_1$, ${\ell }_2=c{Z}_0+d{Z}_1$. It vanishes at exactly one projective point precisely when its coefficient matrix has rank one, equivalently $ad-bc=0$ and the matrix is nonzero. Its kernel is then one line in ${\mathbb{C}}^2$. Equivalently, $s=v\ell$ for a constant nonzero vector $v$ and one nonzero linear form $\ell$. Rank two gives no zero, while rank zero gives the identically zero section.