Counterexamples to Sato's Weak F-Equivalence Conjecture and a Gorenstein Refinement

Abstract

We disprove Sato's weak \(F\)-equivalence conjecture for nonsingular projective toric weak Fano varieties in every dimension \(d \geq 3\). Our counterexamples are smooth projective crepant models of centered reflexive simplices. The key input is a rigidity property of ray polytopes: if \(X_Σ\) is nonsingular and complete and \(-K_{X_Σ}\) is nef, then every nonzero lattice point of \(P_Σ=\operatorname{Conv}(G(Σ))\) is a primitive ray generator. For our models, this rules out every weak-Fano-preserving equivariant blow-up and blow-down throughout the flop class. We then introduce Gorenstein weak \(F\)-equivalence, generated by projective toric birational zigzags through normal projective Gorenstein toric weak Fano varieties, and formulate a corresponding refinement of Sato's conjecture. We prove this refined conjecture in dimensions \(d \leq 3\), as well as for the family of counterexamples constructed above in every dimension. Finally, we show that the refined conjecture implies the inclusion-connectivity of reflexive \(d\)-polytopes modulo unimodular equivalence, which is known for \(d \leq 4\) and remains open for \(d \geq 5\). The results were developed with the assistance of GPT-5.6 Sol.