The symplectic period formula for integrals of wedge products

For a closed complex $1$-form $\alpha$, define a complex singular $1$-cochain $C_{\alpha }$ on each continuous singular path $\sigma$ by the local primitive path integral ${\mathrm{\Pi }}_{\alpha }(\sigma )$. To see that it is a cocycle, take any continuous singular $2$-simplex and subdivide it until each small triangle lies in a neighborhood with a primitive from [F2], using [F3]. The integral around each such triangle is zero because it is the alternating sum of endpoint values of that primitive. Internal edges cancel in opposite orientations, and additivity of path integrals gives $C_{\alpha }(\mathrm{\partial }\sigma )=0$. Thus $C_{\alpha }$ defines a continuous singular cohomology class.

On every smooth singular $1$-simplex, local-primitive increments are the usual integral of the pulled-back form, componentwise on real and imaginary parts. Hence the restriction of $C_{\alpha }$ to smooth singular cochains is the de Rham integration cochain. By the definition of $J_X^1$ in [F4] and the injectivity of the restriction isomorphism there, $[C_{\alpha }]$ is ${\mathcal{J}}_X^1[\alpha ]$. Evaluating it on the fixed side-loop classes gives exactly ${\mathrm{\Pi }}_{\alpha }(a_i),{\mathrm{\Pi }}_{\alpha }(b_i)$; in particular these coordinates depend only on $[\alpha ]$. The construction is complex-linear in $\alpha$: linear combinations of local primitives are local primitives of the same linear combinations of forms.

Put $\theta =\alpha \wedge \beta$. To identify top-degree evaluation with the global integral, cover $X$ by finitely many oriented coordinate rectangles $V_j$ and choose a smooth partition of unity ${\rho }_j$ subordinate to them; full AC supplies the countable-choice hypothesis of that partition supplier. Each ${\theta }_j={\rho }_j\theta$ has compact support in one rectangle. Choose a smaller closed coordinate rectangle $R_j\subset V_j$ whose interior contains that support, and triangulate $R_j$ into two positively oriented affine simplices. Their common edge cancels, and their remaining boundary lies outside $\mathrm{supp}{\theta }_j$, so this relative chain is the positive local orientation generator. The integration cochain of ${\theta }_j$ is a cocycle by the cochain-map supplier and vanishes on simplices in $X\setminus \mathrm{int}(R_j)$. By the smooth-chain comparison it may be evaluated on a smooth representative of $[X]$; the local characterization of $[X]$ and excision identify this evaluation with its evaluation on the rectangle chain. By the finite parametrization formula in [F7], that sum of two simplex integrals is exactly ${\int }_X{\theta }_j$. Sum over $j$ and use ${\sum }_j{\rho }_j=1$ to obtain $⟨{\mathcal{J}}_X^2[\theta ],[X]⟩={\int }_X\theta \mathrm{.}$ This local argument also applies to complex forms by separating real and imaginary parts.

Let $c={\mathcal{J}}_X^1[\alpha ]$ and $d={\mathcal{J}}_X^1[\beta ]$. By [F5], ${\mathcal{J}}_X^2[\alpha \wedge \beta ]=c⌣d$, so step 1.3 gives ${\int }_X\alpha \wedge \beta =⟨c⌣d,[X]⟩\mathrm{.}$ Write $c={\sum }_i(A_i{x}_i+B_i{y}_i)$ and $d={\sum }_i(A_i^{\mathrm{\prime }}{x}_i+B_i^{\mathrm{\prime }}{y}_i)$ in the evaluation-dual complex basis $x_i,y_i$ corresponding to $a_i,b_i$. The matrix in [F6] yields $⟨c⌣d,[X]⟩={\sum }_i(A_i{B}_i^{\mathrm{\prime }}-B_i{A}_i^{\mathrm{\prime }})\mathrm{.}$ By step 1.2, $A_i={\mathrm{\Pi }}_{\alpha }(a_i)$, $B_i={\mathrm{\Pi }}_{\alpha }(b_i)$ and similarly for $\beta$. This proves the wedge-period formula. The matrix $J$ is invertible, and the period-coordinate map is an isomorphism by [F4,F6], so the pairing is nondegenerate. By [F6] it is the stated Poincaré-dual cup pairing.

Replacing $\alpha$ by $\alpha +d\xi$ and $\beta$ by $\beta +d\eta$ changes their wedge by $d(\xi \wedge \beta -\alpha \wedge \eta +\xi \wedge d\eta ),$ so [F8] makes its integral unchanged; step 1.2 also shows that all period coordinates depend only on the classes. Wedge is complex-bilinear and $\alpha \wedge \beta =-\beta \wedge \alpha$, so the descended pairing is complex-bilinear and alternating. If $\omega =h(z)dz$ and $\eta =k(z)dz$ in a local coordinate, then $\omega \wedge \eta =h(z)k(z)dz\wedge dz=0$ on each chart; the formula gives $S(\omega ,\eta )=0$. The period integration lemma in [F10] identifies these holomorphic periods with $P(-,\omega )$, proving the final assertion.