Invariants of ample line bundles on projective varieties and their applications. III
\(i\)-th sectional \(H\)-arithmetic genus
This is the continuation of a series of papers devoted to the study of the geometry of multipolarized manifolds \((X, L_1, \dots , L_{n-i})\), consisting of a smooth complex projective variety \(X\) of dimension \(n\) and ample line bundles \(L_1, \dots , L_{n-i}\) on \(X\), where \(i\) is an integer with \(0 \leq i \leq n-1\). In the previous parts I [Kodai Math. J. 31, No. 2, 219--256 (2008; Zbl 1146.14004)] and II [Kodai Math. J. 33, No. 3, 416--445 (2010; Zbl 1206.14019)], the author introduced and studied general properties of suitable invariants for this investigation, like the \(i\)-th sectional geometric genus \(g_i=g_i(X, L_1, \dots ,L_{n-i})\) and the \(i\)-th sectional \(H\)-arithmetic genus \(\chi^{H}_i= \chi^{H}_i(X, L_1, \dots ,L_{n-i})\). Here, he focuses in particular on the case \(i=2, n\geq 4\), providing some applications of the theory. They include:NEWLINENEWLINE 1) the classification of multipolarized manifolds \((X,L_1, \dots ,L_{n-2})\) with \(g_2=h^2(\mathcal O_X)\), under the assumption that \(L_j\) is spanned for every \(j=1, \dots, n-2\), obtained also by the reviewer in [Rev. Mat. Complut. 13, No.1, 33-48 (2000; Zbl 0993.14003)];NEWLINENEWLINE 2) the proof of appropriate lower bounds for \(\chi^{H}_2\) according to the values of the Iitaka dimension of the adjoint bundle \(K_X+L_1+\dots +L_{n-2}\);NEWLINENEWLINE 3) the explicit computation of all sectional geometric genera of a complete intersection of hypersurfaces in the projective space;NEWLINENEWLINE 4) the notion of \(i\)-th sectional plurigenera for multipolarized manifolds.
scientific article; zbMATH DE number 6073750