Sato’s weak F-equivalence conjecture

Counterexample to Sato's Weak $F$-Equivalence Conjecture

In Counterexamples to Sato's Weak F-Equivalence Conjecture and a Gorenstein Refinement, Chakravarty, Choi, and Xu [CCX26] disprove Sato's weak $F$-equivalence conjecture in every dimension $d \ge 3$.

The counterexamples are constructed from the centered reflexive simplex. For every $d\ge 3$, let

$$N_d = \left\{ (b_0,\ldots,b_d)\in \mathbb Z^{d+1} : \sum_{i=0}^d b_i=0 \right\},$$

and define

$$\Delta_d = \left\{ (b_0,\ldots,b_d)\in (N_d)_{\mathbb R} : b_i\ge -1 \right\}.$$

Equivalently,

$$\Delta_d = \operatorname{Conv}(v_0,\ldots,v_d), \qquad v_i=(d+1)e_i-\mathbf 1.$$

Choose a coherent unimodular triangulation of $\partial\Delta_d$ using all of its lattice points. This produces a smooth projective crepant toric resolution

$$X_d\longrightarrow X_{\Delta_d}.$$

The primitive ray generators of the fan of $X_d$ are exactly the nonzero lattice points of $\Delta_d$:

$$G(\Sigma_d) = (\Delta_d\cap N_d)\setminus\{0\}.$$

Since

$$\#(\Delta_d\cap N_d) = \binom{2d+1}{d},$$

the fan of $X_d$ has

$$r_d = \binom{2d+1}{d}-1$$

rays.

Why this gives a counterexample

The key rigidity result is that if $X_\Sigma$ is nonsingular and complete and $-K_{X_\Sigma}$ is nef, then

$$\operatorname{Conv}(G(\Sigma))\cap N = \{0\}\cup G(\Sigma).$$

Thus every nonzero lattice point of the ray polytope is already a primitive ray generator.

For the varieties $X_d$, this rigidity prevents any nontrivial toric blow-up or blow-down from remaining inside the class of smooth toric weak Fano varieties. Toric flops, on the other hand, preserve the ray set.

Consequently, every smooth toric weak Fano variety weakly $F$-equivalent to $X_d$ has exactly

$$\binom{2d+1}{d}-1$$

rays.

However, the fan of $\mathbb P^d$ has only

$$d+1$$

rays. Therefore

$$X_d\not\sim_{\mathrm{wF}}\mathbb P^d \qquad\text{for every }d\ge 3.$$

Hence $X_d$ gives a counterexample to Sato's weak $F$-equivalence conjecture in every dimension $d\ge 3$.

The first counterexample: dimension $3$

In dimension three,

$$N_3 = \left\{ (b_0,b_1,b_2,b_3)\in\mathbb Z^4 : b_0+b_1+b_2+b_3=0 \right\},$$

and

$$\Delta_3 = \operatorname{Conv} \left\{ \begin{aligned} &(3,-1,-1,-1),\\ &(-1,3,-1,-1),\\ &(-1,-1,3,-1),\\ &(-1,-1,-1,3) \end{aligned} \right\}.$$

It contains

$$\binom{7}{3}=35$$

lattice points. The origin is the unique interior lattice point, so the remaining $34$ nonzero lattice points become the $34$ rays of a smooth projective crepant model $X_3$.

Thus

$$\#G(\Sigma_{X_3})=34,$$

whereas

$$\#G(\Sigma_{\mathbb P^3})=4.$$

Since the ray set of $X_3$ cannot change under smooth weak-Fano-preserving blow-ups, blow-downs, or flops, one has

$$X_3\not\sim_{\mathrm{wF}}\mathbb P^3.$$

This is the basic three-dimensional counterexample, and the same construction extends uniformly to every dimension $d\ge3$.