Volume of Algebraically Integrable Foliations and Locally Stable Families

The paper proves that for any lc algebraically integrable foliated triple (X, F, B) of fixed rank d, with coefficients in a DCC set Γ and fixed leaf volume v, there exists a uniform positive lower bound δ for vol(KF+B) when KF+B is big, and moreover the possible volumes lie in a discrete set depending only on d, Γ, v (Theorem 1.5). Their proof reduces, via ACSS/core modifications and finite covers, to a stable family and shows vol(KF+B) = l·vol(KXT/T+BT), where dim XT is uniformly bounded in terms of d, Γ, v, whence discreteness and a uniform lower bound follow. The candidate’s product construction ignores this moduli-theoretic reduction. In particular, their isotrivial product with a horizontal boundary cannot survive the reduction to a (generically finite) moduli image without violating Lemma 2.19 (which forbids bigness for stable families not of maximal variation). Further, the candidate mis-normalizes volume (vol(D)=D^n, not D^n/n!) and omits the binomial factor in the product self-intersection; although these errors do not salvage their conclusion numerically, the decisive issue is that, after the paper’s reduction steps, vol(KF+B) is controlled by intersection numbers on a space of bounded dimension with bounded indices, so it cannot tend to 0 nor accumulate in the manner proposed. Thus the paper’s statements (1) and (2) stand, and the model’s counterexamples fail.