A note on the dimension of global sections of adjoint bundles for polarized 4-folds
Let \((X,L)\) be a polarized manifold of dimension \(n\) over the complex field, where \(L\) is ample. The author of the article under review is interested in establishing a lower bound on \(h^0(K_X+tL)\) for \(t>0\). To this end, in a previous paper he introduced an invariant \(A_i(X,L)\), for any \(0\leq i\leq n\), called \(i\)-th Hilbert coefficient of \((X,L)\). For a fixed \(t\), it is defined as follows. First define inductively \(F_0(t)=h^0(K_X+tL)\), and \(F_i(t)=F_{i-1}(t+1)-F_{i-1}(t)\). Then define \(A_i(X,L)=F_{n-i}(1)\). Here the case \(n=4\) is considered. It is proved that \(A_2(X,L)\geq 0\). Moreover, if \(m\geq 3\) it is proved that \(h^0(K_X+mL)\geq h^0(K_X+(m-1)L)\). A geometrical description is also given of the \(4\)-folds possibly failing the inequality in the case \(m=2\). This answers to a question posed by H. Tsuji.
scientific article; zbMATH DE number 6941822