Irrationality is not the obstruction: support property for Toda's Gepner heart on the quintic threefold

Abstract

I show that the irrationality of the coefficients in Toda's conjectural Gepner-type central charge Z†G on the smooth quintic threefold X ⊂ P⁴ is not a fundamental obstruction to the existence of Harder--Narasimhan filtrations in his proposed heart A_G. Using only the linearity of Z†G on K₀(X), the structure of the first tilt B{B,H} with B=-H/2, and Xu's 2025 Bogomolov--Gieseker inequality for sheaves with slope -H/2, I prove analytically that ν_G-semistable objects with ν_G(E) in a bounded interval have Chern characters in a bounded region of K₀(X)⊗R. Combined with the discreteness of K₀(X) ≅ Z⁴, this yields the support property required for HN filtrations via Bridgeland's criterion. I also verify computationally that Li's family σ{α,β,H} does not contain a Gepner point, and analyze why the simplified model of ν_G-semistable objects as complexes F→T with both components having slope -H/2 fails. My results clarify that the remaining obstacle to Toda's Conjecture 3.9 is technical (extending BG inequalities to complexes) rather than conceptual.

Keywords