Big \(q\)-ample line bundles (Q2894203)

According to [\textit{B. Totaro}, ``Line bundles with partially vanishing cohomology'', \url{arXiv:1007.3955}], a line bundle \(L\) on a complex projective variety \(X\) is \(q\)-ample (for a non-negative integer \(q\)) if for every coherent sheaf \(F\) on \(X\) there exists an integer \(m_0\) depending on \(F\) such that \(H^i(X,F(mL))=0\) for all \(m\geq m_0\) and all \(i>q\). This is equivalent to a variant introduced in [\textit{J.-P. Demailly, T. Peternell} and \textit{M. Schneider}, Isr. Math. Conf. Proc. 9, 165--198 (1996; Zbl 0859.14005)]. This is a subtle notion, and the only well-understood cases are \(0\)-ampleness (which is just the usual ampleness) and \((n-1)\)-ampleness.NEWLINENEWLINEIf \(L\) is a big Cartier divisor on \(X\), then \(\mathbf{B}_+(X)\) is defined as the intersection of stable base loci \(\mathbf{B}(L-\varepsilon A)\) for a fixed ample divisor \(A\) and for all \(\varepsilon>0\). Then the main result of this paper is that \(L\) is \(q\)-ample if and only if \(L_{|Y}\) is \(q\)-ample, where \(Y\) is \(\mathbf{B}_+(L)\) equipped with the reduced scheme structure.