Hermitian metric and $L^2$ pairing on a compact Riemann surface

If $z_j=\varphi (z_i)$ is a holomorphic change of coordinate, its real Jacobian is multiplication by the complex derivative ${\varphi }^{\mathrm{\prime }}$, so it commutes with multiplication by $i$ and has determinant $\mathrm{\mid }{\varphi }^{\mathrm{\prime }}{\mathrm{\mid }}^2>0$. Thus the local rotations defining $J$ agree on overlaps and the charts orient $X$ consistently. The complexified cotangent eigenbundles are consequently well-defined, with ${\mathrm{\Lambda }}^{0,1}{T}^{\ast }X=\bar{K}$ and ${\mathrm{\Lambda }}^{0,0}{T}^{\ast }X$ trivial.

By [F2] choose a Riemannian metric $g_0$ and average it with $J$; the result is positive and satisfies $g(Jv,Jw)=g(v,w)$. In a chart, writing $g=ad{x}^2+2bdxdy+cd{y}^2$, this identity gives $a=c$ and $b=0$, hence $g=\rho (d{x}^2+d{y}^2)$ with $\rho =a>0$. Its determinant is ${\rho }^2$, so the Riemannian density and complex-oriented volume form have the displayed local formulas. The determinant metric on covectors makes $dz,d\bar{z}$ orthogonal with squared norms $2\mathrm{/}\rho$, establishing the type orthogonality.

By [F2] the underlying real bundle of $E$ has a smooth positive real fibre metric $b$. Its average $b_J(v,w)=\frac{1}{2}(b(v,w)+b(J_{E}v,J_{E}w))$ is smooth, positive and $J_E$-invariant. Invariance makes $J_E$ orthogonal and skew-adjoint, so $b_J(J_{E}v,v)=0$; direct substitution then shows that $h(v,w)=b_J(v,w)-i{b}_J(J_{E}v,w)$ is complex-linear in $v$, conjugate-linear in $w$, Hermitian, and satisfies $h(v,v)=b_J(v,v)>0$ for $v\mathrm{\ne }0$. Hence it is a smooth Hermitian bundle metric, and a holomorphic frame has smooth positive weight $\psi$.

In a local frame the Hermitian dual identification sends $e$ to $\psi e^{\ast }$. Combining it with the conjugate-linear metric star on complex forms gives the two displayed local formulas for ${\star }_E$. For $q=0$, $fe\wedge {\star }_E(ge)=\psi f\bar{g}d{V}_g$. For $q=1$, $d\bar{z}\wedge dz=2idx\wedge dy$ and $⟨d\bar{z},d\bar{z}{⟩}_g=2\mathrm{/}\rho$, so $fd\bar{z}\otimes e\wedge {\star }_E(ud\bar{z}\otimes e)=(2\psi f\bar{u})dx\wedge dy=⟨fd\bar{z}\otimes e,ud\bar{z}\otimes e⟩d{V}_g$. The defining pairing is nondegenerate, so these formulas determine a unique global conjugate-linear map with target bidegree $(1,1-q)$. For $adz\otimes e^{\ast }$, the same calculation gives ${\star }_{E^{\ast }}(adz\otimes e^{\ast })=i{\psi }^{-1}\bar{a}d\bar{z}\otimes e^{\ast \ast }$; hence ${\star }_{E^{\ast }}{\star }_E=-1$ on $(0,1)$-forms and ${\star }_E^{-1}=-{\star }_{E^{\ast }}$.

For compactly supported smooth $s,t$, the pointwise pairing is smooth and the volume density is positive and smooth, so its integral is finite; linearity and conjugate symmetry follow pointwise and from the integral. If $s\mathrm{\ne }0$, its pointwise squared norm is positive on a neighborhood of a point where $s$ is nonzero, so the integral is strictly positive. The defining equation for ${\star }_E$ yields $⟨s,t{⟩}_{L^2}={\int }_{X}s\wedge {\star }_{E}t$. Pointwise Cauchy–Schwarz [F7] gives $\mathrm{\mid }⟨s,t⟩\mathrm{\mid }\le \mathrm{\mid }s\mathrm{\mid }\mathrm{\mid }t\mathrm{\mid }$, and scalar $L^2$ Cauchy–Schwarz [F5] then gives $\mathrm{\mid }⟨s,t{⟩}_{L^2}\mathrm{\mid }\le \mathrm{\parallel }s{\mathrm{\parallel }}_{L^2}\mathrm{\parallel }t{\mathrm{\parallel }}_{L^2}$.

Choose a finite holomorphic-chart/frame cover and a subordinate smooth partition $({\chi }_j)$. For a measurable square-integrable section $s$, each ${\chi }_{j}s$ has compact support $K_j\subset U_j$. In the chart, its coefficient extended by zero lies in Euclidean $L^2$. By [F8], choose a cutoff ${\eta }_j$ supported in $U_j$ and equal to one near $K_j$. On the compact support of ${\eta }_j$, the smooth positive metric and volume weights are bounded above and below. Approximate the coefficient by Euclidean $C_c^{\mathrm{\infty }}$ functions from [F5], multiply those approximants by ${\eta }_j$, convert them back to sections, and extend by zero; the norm equivalence on $\mathrm{supp}({\eta }_j)$ makes the resulting sections converge to ${\chi }_{j}s$. Summing over the finite cover proves density of compactly supported smooth sections in the measurable realization. The countable-choice completion interface in [F5] makes the completion a complex Hilbert space. The only choice assumption is ${\mathrm{A}\mathrm{C}}_{\omega }$ through the metric, partition, density-integral and completion suppliers listed in [F2], [F4], and [F5].

The completion of the dense inner-product space is the stated $L^2$ Hilbert space, and its norm is the completion of the bundle norm. The smooth compactly supported forms are dense by construction and by step 3.1, so the two descriptions agree.