= Holomorphic projectivization by lines
{title2=$\mathbb P(E)_x=\{\ell\subset E_x:\dim\ell=1\}$}
For a <holomorphic vector bundle>, projectivize each fibre using lines. The transition $g_{ij}(x)$ acts holomorphically by $[v]\mapsto[g_{ij}(x)v]$. Thus local products $U_i\times\mathbb P^{r-1}$ define a <complex manifold> with holomorphic projection to the base. The lines convention gives the <relative tautological line bundle> of fibre degree $-1$, and its dual has degree $+1$.
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