On volumes along subvarieties of line bundles with nonnegative Kodaira-Iitaka dimension (Q538006)

Let \(X\) be a smooth projective variety and \(L\) a line bundle on \(X\) with nonnegative Kodaira-Iitaka dimension \(\kappa(L)\geq 0\). Let \(V\subset X\) be a subvariety of dimension \(d>0\) such that \(V\) is not contained in \(SBs(L)\), where \(SBs(L)= \bigcap_{m>0}Bs|mL|\) is the stable base locus. Let \[ H^0(X|V, mL)={\text{Image}}[H^0(X, mL)\rightarrow H^0(V, mL)] \] be the image of restriction maps. The restricted volume of \(L\) along \(V\) is defined to be \[ {\text{vol}}_{X|V}(L)=\limsup_{m\rightarrow \infty} \frac{h^0(X|V, mL)}{m^d/d!}. \] The reduced volume of \(L\) along \(V\) is \[ \mu(V, L)=\limsup_{m\rightarrow \infty} \frac{h^0(V, {\mathcal{O}}_V( mL)\otimes {\mathcal{I}}(\|mL\| )|_V)} {m^d/d!}, \] where \({\mathcal{I}}(\|mL\|)={\mathcal{I}}(X, \|mL\|)\) is the asymptotic multiplier ideal sheaf of \(mL\) for every positive integer \(m\). In this paper, the authors prove that both \[ \limsup_{m\rightarrow \infty} \frac{h^0(X|V, mL)}{m^q} \] and \[ \limsup_{m\rightarrow \infty} \frac{h^0(V, {\mathcal{O}}_V( mL)\otimes {\mathcal{I}}(\|mL\| )|_V)} {m^q} \] are finite positive numbers if \(V\) contains a general point of \(X\) and \(\dim f(V)=q\geq 0\), where \(f:X\rightarrow Y\) is the Iitaka fibration associated to \(L\). They also prove that for a very general point \(x\in X\), if \(\mu(C, L)= {\text{vol}}_{X|C}(L)\) for any curve \(C\) passing through \(x\), then either \(\kappa (L)=0\) or \(\kappa (L)=\)dim\(X\). Their proofs rely on techniques of multiplier ideal sheaves and dimension counting arguments.