Calabi–Yau threefold intersection and Kähler-volume coordinates

Let $X$ be the smooth compact Kähler threefold of the compactification datum, and use its complex orientation for every integral and duality convention below. Choose a real basis $[{\omega }_a]$, $1\le a\le h^{1,1}(X)$, of $H^{1,1}(X;\mathbb{R})$, represented by smooth real closed $(1,1)$-forms. When this space is spanned by integral classes, the basis may be chosen integral and its lattice normalization must be stated; broad $K_X$-triviality alone does not imply that such an integral basis spans the whole real space. For the dimensionless Kähler form $J$ in ${\alpha }^{\mathrm{\prime }}=1$ units, write its Kähler class as $[J]=v^a[{\omega }_a]$ with unique real coefficients $v^a$. No equality between the selected form representatives is imposed. Define ${\kappa }_{abc}={\int }_X{\omega }_a\wedge {\omega }_b\wedge {\omega }_c$, $\mathcal{V}=\frac{1}{6}{\int }_X{J}^3=\frac{1}{6}{\kappa }_{abc}{v}^a{v}^b{v}^c$, and ${\tau }_a=\mathrm{\partial }\mathcal{V}\mathrm{/}\mathrm{\partial }{v}^a=\frac{1}{2}{\kappa }_{abc}{v}^b{v}^c$. Thus $v^a$ are Kähler-cone coordinates; if a dual two-cycle basis ${\gamma }^a$ satisfies ${\int }_{{\gamma }^a}{\omega }_b={\delta }_b^a$, then $v^a={\int }_{{\gamma }^a}J$ and are curve volumes when those classes are effective. If a divisor basis $D_a$ is supplied with each $D_a$ dual to $[{\omega }_a]$, the associated divisor-volume coordinate is defined in this intersection convention by $\tau (D_a):=\frac{1}{2}{\kappa }_{abc}{v}^b{v}^c={\tau }_a$; without such a divisor basis, ${\tau }_a$ are the dual real coordinates defined by $\mathrm{\partial }\mathcal{V}\mathrm{/}\mathrm{\partial }{v}^a$. The total dimensionless volume is $\mathcal{V}$, and the physical volume is $({\alpha }^{\mathrm{\prime }}{)}^3\mathcal{V}$ in this convention. For Type IIB O3/O7 data, indices are restricted to a specified orientifold-even Kähler basis and odd-basis intersection couplings are stated separately.