The period pairing is well defined and computed by integration

Let $\sigma :{\mathrm{\Delta }}^2\to X$ be any continuous singular $2$-simplex. The coordinate disks equipped with local primitives from [F4] form an open cover of $X$. By [F5], some iterated barycentric subdivision of $\sigma$ is a finite sum of small singular $2$-simplices, each mapped into a disk carrying a primitive $H$. For one such triangle $\tau$, [F2] gives ${\int }_{\tau {\delta }_0}\omega =H(\tau (v_2))-H(\tau (v_1))$, ${\int }_{\tau {\delta }_1}\omega =H(\tau (v_2))-H(\tau (v_0))$, and ${\int }_{\tau {\delta }_2}\omega =H(\tau (v_1))-H(\tau (v_0))$; the alternating face sum is zero. Summing over the subdivision cancels interior faces in opposite orientations, and additivity of the edge path integrals recovers the original boundary. Hence ${\int }_{\mathrm{\partial }\sigma }\omega =0$.

Extend $C_{\omega }(\sigma ):={\int }_{\sigma }\omega$ from continuous singular $1$-simplices to the complex singular $1$-cochain by finite linearity. Applying step 1.1 to each simplex in a finite $2$-chain $\beta$ gives $C_{\omega }(\mathrm{\partial }\beta )={\int }_{\mathrm{\partial }\beta }\omega =0$. Thus $C_{\omega }$ is a cocycle and its evaluation on a $1$-cycle depends only on that cycle's homology class; it equals the complex-coefficient Kronecker evaluation of $[C_{\omega }]$ on that class.

Let $\gamma$ have coordinates ${\sum }_i(m_i{a}_i+n_i{b}_i)$ in the basis of [F1], and let $c$ be any continuous singular cycle representing it. The cocycle from step 2.1 evaluates on $a_i,b_i$ as the path integrals over their fixed representatives $A_i,B_i$. By [F1] these are exactly the defining values used in $P$; additivity and the basis expansion therefore give $C_{\omega }(c)=P(\gamma ,\omega )$. If $c$ is piecewise $C^1$, [F2] identifies this value with the usual contour integral. The same canonical homology functional is obtained from any symplectic basis or representative.

Additivity of the path integral in the chain and its $\mathbb{C}$-linearity in $\omega$ from [F2] imply the stated bilinearity of $P$ by step 3.1. If $v_i^{\mathrm{\prime }}={\sum }_j{U}_{ij}{v}_j$, then $P(v_i^{\mathrm{\prime }},\omega )={\sum }_j{U}_{ij}P(v_j,\omega )$ for each $i$, so $p^{\mathrm{\prime }}=Up$. For $g=0$ both vectors are empty.

Write $\omega =\alpha +i\beta$ as in [F11]. By [F3], $\alpha$ and $\beta$ are closed real $1$-forms. The real and imaginary parts of $C_{\omega }$ from step 2.1 are continuous singular $1$-cocycles. By [F9], $X$ is a finite-dimensional Hausdorff second-countable smooth manifold, so [F7] applies. On every smooth singular $1$-simplex, [F2] and [F7] identify the cocycle restrictions with the de Rham integration cochains $I_X(\alpha )$ and $I_X(\beta )$. Thus $r_X[\mathrm{Re}{C}_{\omega }]=I_X[\alpha ]$ and $r_X[\mathrm{Im}{C}_{\omega }]=I_X[\beta ]$; injectivity of $r_X$ in [F7] gives $[\mathrm{Re}{C}_{\omega }]=J_X[\alpha ]$ and $[\mathrm{Im}{C}_{\omega }]=J_X[\beta ]$. Evaluating by [F8] on $\gamma$ and using step 3.1 yields the displayed de Rham identity. Full AC supplies the symplectic basis in [F1] and implies ${\mathrm{A}\mathrm{C}}_{\omega }$ through [F10]; the local subdivision and endpoint calculations use only finite choice.