The Laurent-coefficient 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 $x_h$. Let $A_2(\zeta_R)$ and $A_3(\zeta_R)$ be the corresponding coefficients in the Laurent expansion of the Hilbert series. Laurent-coefficient Futaki bound.

$$F(\mathcal{X};\zeta_R,\eta_h)\geq \frac{3}{8}\bigl(A_2(\zeta_R)-A_3(\zeta_R)\bigr).$$

The inequality is established empirically for the 16 reflexive polygons and conjectured for all toric Calabi–Yau threefolds without a factor of $\mathbb{C}$; its general validity remains open.