On complex manifolds polarized by an ample line bundle of sectional genus \(q(X)+2\)
Let \(X\) be a smooth projective variety of dimension \(n\) over the complex field and \(L\) be an ample line bundle on \(X\): Then the pair \((X,L)\) is called a polarized manifold. By definition, \(g(L)=1+{1\over 2}(K_X+(n-1)L)L^{n-1}\) is the sectional genus and \(q(X)=h^1({\mathcal O}_X)\) is the irregularity. Fujita conjectured that, in this situation, \(g(L)\geq q(X)\). \textit{Y. Fukuma} [Trans. Am. Math. Soc. 351, No. 1, 363-377 (1999; Zbl 0905.14003)], proved the conjecture for threefolds with \(h^0(L)\geq 2\) under the weaker assumption that \(L\) is nef and big. Moreover he gave a complete classification of polarized threefolds with \(g(L)=q(X)\) and \(h^0(L)\geq 3\), or with \(g(L)=q(X)+1\) and \(h^0(L)\geq 4\). In the paper under review, the author studies the subsequent case of polarized threefolds with \(g(L)=q(X)+2\) and \(h^0(L)\geq 5\). He proves that there are only two possibilities, i.e. \((X,L)\) is either a hyperquadric fibration over \(\mathbb P^1\) or a scroll over a smooth surface \(Y\) with \(q(Y)=0\). The second main result of the paper is the complete classification of polarized manifolds of dimension \(n\geq 3\) with \(g(L)=q(X)+2\) and \(\text{Bs}|L|=\emptyset\).
scientific article; zbMATH DE number 1502090