Title: On volumes along subvarieties of line bundles with nonnegative Kodaira-Iitaka dimension
Abstract:We study the restricted volume along subvarieties of line bundles with non-negative Kodaira-Iitaka dimension. Our main interest is to compare it with a similar notion defined in terms of the asymptoti...We study the restricted volume along subvarieties of line bundles with non-negative Kodaira-Iitaka dimension. Our main interest is to compare it with a similar notion defined in terms of the asymptotic multiplier ideal sheaf, with which it coincides in the big case. We shall prove that the former is non-zero if and only if the latter is. We then study inequalities between them and prove that if they coincide on every very general curve the line bundle must have zero Kodaira-Iitaka dimension or be big. Let X be a smooth projective variety, L a divisor or a line bundle on X with nonnegative Kodaira-Iitaka dimension: κ(L) ≥ 0. Let V ⊂ X be a subvariety of dimV = d > 0 such that V 6⊂ SBs (L), where SBs (L) := ⋂ m>0 Bs |mL| is the stable base locus. We denote by H(X|V,mL) = Image [H(X,mL) −→ H(V,mL)] the image of restriction maps. The restricted volume of L along V is defined to beRead More