Selectional geometric genera for ample vector bundles (Q1884369)

Inspired by some work of \textit{T. Fujita} [J. Math. Kyoto Univ. 29, 1--16 (1989; Zbl 0699.14043)], \textit{A. Lanteri} [Rev. Mat. Comput. 13, 33--48 (2000; Zbl 0993.14003)] and \textit{Y. Fukuma} [Commun. Algebra 32, 1069--1100 (2004; Zbl 1068.14008)], the author introduces and studies two numerical invariants associated to a pair \((X,E)\) consisting of a projective manifold \(X\) and an ample vector bundle \(E\) on \(X\), namely the numerical invariants \(g_i(X,\det(E))\) and \(g_i(\mathbb P(E),\mathcal O_{\mathbb P(E)}(1))\) (called \(c_1\)-sectional genera and \(\mathcal O(1)\)-sectional genera) associated to the polarized varieties \((X,\det(E))\) and \((\mathbb P(E),\mathcal O_{\mathbb P(E)}(1))\) respectively. When \(E\) is spanned these two invariants are non-negative, in fact they are \(\geq h^i(\mathcal O_X)\). Finally, the author classifies all pairs \((X,E)\) as above such that \(g_2(X,\det(E))=h^2(\mathcal O_X)\), or \(g_2(\mathbb P(E),\mathcal O_{\mathbb P(E)}(1))=h^2(\mathcal O_X)\).