The harmonic-forms and intersection-cohomology conjecture for F-theory

Let $\mathcal{B}$ be the base of an elliptically fibered Calabi–Yau manifold $pi:X\tomathcal{B}$, with discriminant divisor $\Delta$, and set $U=\mathcal{B}\setminusDelta$. Let $\mathbb{V}=R^1(\pi_U)_*\mathbb{R}$ be the local system on $U$. For an integer $i$, consider $\mathbb{V}$-valued $i$-forms on $U$.

Harmonic intersection-cohomology conjecture. The vector space of $\mathbb{V}$-valued $i$-forms that are harmonic and $L^2$ with respect to the Kähler metric is isomorphic to

$$\mathbb{H}^i(\mathcal{B},IC(\mathbb{V})).$$

The conjecture is proposed for F-theory compactifications associated to elliptically fibered Calabi–Yau manifolds in any dimension. The source presents the assertion as a conjectural extension of the known one-dimensional-base situation.