Every holomorphic line bundle on a compact Riemann surface has a meromorphic section

By [F1], choose compatible metrics on $X$ and $E$ to apply [F3], and set $m={\dim }_{\mathbb{C}}{H}^{0,1}(X,E)<\mathrm{\infty }$. Choose a holomorphic coordinate disk $U$ about $p$ with $z(p)=0$ and a holomorphic frame $e$ for $E$ on $U$. Choose a smaller closed coordinate disk $K_0\subset U$ whose interior contains $p$, and by [F7] choose $\chi \in C_c^{\mathrm{\infty }}(U)$ equal to $1$ near $K_0$; its support is compact because $X$ is compact.

For each $j=1,\dots ,m+1$, define ${\sigma }_j=\chi z^{-j}e$ on $U\setminus \{p\}$ and extend it by zero to $X\setminus \{p\}$. This is smooth away from $p$ because $\chi$ is supported inside $U$. Define ${\theta }_j={\bar{\mathrm{\partial }}}_E{\sigma }_j$ on $X\setminus \{p\}$. On a neighborhood of $p$ one has $\chi =1$, so ${\sigma }_j=z^{-j}e$ there and ${\theta }_j=0$; thus ${\theta }_j$ extends by zero to a global smooth $E$-valued $(0,1)$-form. Every such form is ${\bar{\mathrm{\partial }}}_E$-closed because $(0,2)=0$ by [F8].

Define $T:{\mathbb{C}}^{m+1}\to H^{0,1}(X,E)$ by $T(c)=[{\sum }_{j=1}^{m+1}{c}_j{\theta }_j]$. This is complex-linear by [F2]. Choose a basis of its $m$-dimensional target; the equation $T(c)=0$ then has $m$ homogeneous linear equations in $m+1$ unknowns. Row reduction has at most $m$ pivots and leaves a free variable, so choose $c\mathrm{\ne }0$ in the kernel, including when $m=0$. Put $\theta ={\sum }_j{c}_j{\theta }_j$; its Dolbeault class is zero.

Since $[\theta ]=0$, the quotient definition [F2] supplies a global smooth section $u$ of $E$ with ${\bar{\mathrm{\partial }}}_{E}u=\theta$. The only choice hypothesis is inherited through metric existence and finite-dimensional cohomology; the kernel calculation is finite.

On $X\setminus \{p\}$ set $s={\sum }_j{c}_j{\sigma }_j-u$. Then ${\bar{\mathrm{\partial }}}_{E}s=0$, so $s$ is holomorphic there by [F4]. Near $p$, where $\chi =1$, its coefficient in the frame $e$ is ${\sum }_j{c}_j{z}^{-j}-u_e(z)$. Since $\theta =0$ near $p$, [F4] makes $u_e$ holomorphic there; the nonzero Laurent polynomial ${\sum }_j{c}_j{z}^{-j}$ has a pole, so this coefficient has the same nonzero principal part. Thus $s$ extends meromorphically across $p$, has a pole there, is holomorphic elsewhere, and is not identically zero.