Implication from IT to generic vanishing or M-regularity in the original generic vanishing conjecture

Establish that condition (3), namely that the ideal sheaf I_X(2H) satisfies IT_0, implies condition (2), that I_X(H) is a GV-sheaf, or condition (4), that O_X(H) is M-regular with χ(X,O_X(H))=1, for geometrically nondegenerate closed reduced subschemes X of pure dimension 1≤d≤g−2 in indecomposable principally polarized abelian varieties.

Background

The paper recalls the generic vanishing conjecture proposed by Pareschi and Popa for an indecomposable principally polarized abelian variety (A,H) of dimension g and a geometrically nondegenerate closed reduced subscheme X of pure dimension 1≤d≤g−2. Among its equivalent conditions are the IT_0 property of I_X(2H), the GV property of I_X(H), and M-regularity of O_X(H) together with Euler characteristic one.

The paper notes that several implications in the conjecture are known, including (2)⇒(1) and (2)⇔(4)⇒(3), but the reverse implication from the IT_0 condition (3) to the GV or M-regularity conditions remains unresolved. The present paper proves the analogous implication for effective divisors, i.e. the codimension-one case, rather than resolving the original conjecture for higher-codimension subschemes.