Multiplicity bounds for \d835\dcb-Gorenstein toric varieties of rank r

Let $X$ be an $n$-dimensional $\delta 835\dcb$-Gorenstein toric variety of rank $r$ and Gorenstein index $k\ge 2$. Let $Q^\circ$ be the polar weight matrix defined in the source, and set $r^\circ:=\operatorname{rk} Q^\circ$ and $r':=\max(r,r^\circ)$. Let $t_{k,n}$ be the recursively defined sequence from the source, and let $\mu_{n,r'}$ denote the McMullen number. Multiplicity-bound conjecture.

$$\operatorname{mult} X\le \left[\frac{2k(k+1)^2}{2+r'}\right]\quad\text{if }n=2,$$ $$\operatorname{mult} X\le \left[\frac{4t_{k,3}^2}{(7+r')k}\right]=\left[\frac{4k(k+1)^2\bigl[k(k+1)+1\bigr]^2}{7+r'}\right]\quad\text{if }n=3,$$

and

$$\operatorname{mult} X\le \left[\frac{2t_{k,n}^2k^n}{k\mu_{n,r'}}\right]\quad\text{if }n\ge 4.$$

These bounds are proposed as an extension of the corresponding bounds for fake weighted projective spaces from the preceding theorem to arbitrary $\mathbb{Q}$-Gorenstein toric varieties. The source presents them as plausible consequences of an extension of the cited anti-canonical self-intersection bounds, so their resolution is not established here.