The divisor-volume lower bound for Futaki invariants

Let $\mathcal{X}$ be a toric Calabi–Yau $3$-fold with no factors of $\mathbb{C}$, let $\zeta_R$ be the Reeb symmetry, and let $\eta_h$ be the test symmetry associated with a generator $x_h$. For a toric divisor $\Sigma_\alpha$, let $V(b^*;\Sigma_\alpha)$ denote its volume at the minimizing Reeb vector. Divisor-volume Futaki bound.

$$F(\mathcal{X};\zeta_R,\eta_h)\geq \frac{27}{64}V(b^*;\Sigma_\alpha).$$

This bound is observed for the 16 reflexive polygons and is proposed for all toric Calabi–Yau threefolds without a factor of $\mathbb{C}$; the general case remains open.