On the sectional geometric genus of quasi-polarized varieties. II
Let \(X\) be a projective variety of dimension \(n\) over the complex number field, and let \(L\) be a nef and big line bundle on \(X\). Then one says that the pair \((X,L)\) is a quasi-polarized variety. In a previous paper [Commun. Algebra 32, 1069-1100 (2004; Zbl 1068.14008)], the author introduced the notion of \(i\)-th sectional geometric genus \(g_i(X,L)\) of a quasi-polarized variety \((X,L)\). When \(X\) is smooth and the linear system \(| L| \) is base point free, then \(g_i(X,L)\) is the geometric genus of the transversal intersection \(X_{n-i}\) of general \(n-i\) members of \(| L| \). This accounts for the reason and the interest in this invariant \(g_i(X,L)\). In the quoted paper, the author studied some fundamental properties of the sectional geometric genus, and generalized some problems about the sectional genus \(g_1(X,L)\) to the case of the \(i\)-th sectional geometric genus, \(i\geq 1\). For instance, starting from \textit{T. Fujita's} conjecture [``Classification Theories of Polarized Varieties'' (Lond. Math. Soc. Lect. Note Ser. 155) (1990; Zbl 0743.14004)], he proposed the following conjecture: \[ g_i(X,L)\geq h^i(\mathcal O_X) \] for any \(0\leq i\leq n\). The author proved this conjecture when \(Bs| L| =\emptyset\). Continuing this earlier work, in the paper under review the author studies some properties of the sectional geometric genus, and analyzes previous conjecture for some special varieties, like abelian, Calabi-Yau and Fano varieties, and proves it when \(X\) is an abelian variety and \(L\) is ample.
scientific article; zbMATH DE number 2078009