The Euler-characteristic lower-bound conjecture for reflexive toric Calabi–Yau cones

Let ${\cal X}$ be a toric Calabi–Yau $n$-fold with toric diagram $\Delta_{n-1}$, let $Y$ be its Sasaki–Einstein base, and let $V(b_i^{*};Y)$ denote the minimum volume. Let $\widetilde{X(\Delta_{n-1})}$ be the completely resolved toric variety associated with $\Delta_{n-1}$, and let $\chi(\widetilde{X(\Delta_{n-1})})$ be its Euler number. Euler-characteristic lower-bound conjecture. The minimum volume satisfies

$$V(b_i^{*};Y)\geq \frac{1}{\chi(\widetilde{X(\Delta_{n-1})})}.$$

The bound is saturated when ${\cal X}$ is an Abelian orbifold of $\mathbb{C}^{n}$. The claim is motivated by computations for reflexive toric Calabi–Yau $3$-, $4$-, and $5$-folds; its asserted universality for all reflexive toric Calabi–Yau $n$-folds remains open.