Classification of polarized manifolds by the second sectional Betti number (Q383992)

Let \(X\) be a smooth projective variety of dimension \(n\) over the field of complex numbers \(\mathbb C\) and let \(L\) be an ample line bundle on \(X\). Then the pair \((X,L)\) is called a polarized manifold.NEWLINENEWLINEIn [J. Pure Appl. Algebra 209, No. 1, 99--117 (2007; Zbl 1106.14003)], the author of the paper under review defined the \textit{\(i\)-th sectional Euler number} \(e_i(X,L)\) and the \textit{\(i\)-th sectional Betti number} \(b_i(X,L)\) for each integer \(i\), and proved \(b_i(X,L) \geq h^i(X, \mathbb{C})\) when \(L\) is globally generated. It is then natural to consider the problem of classifying polarized manifolds \((X,L)\) such \(L\) is globally generated and the value of \(b_i(X,L)-h^i(X, \mathbb{C})\) is small. In [Adv. Geom. 8, No.4, 591--614 (2008; Zbl 1156.14007)], he classified polarized manifolds \((X,L)\) such that \(L\) is globally generated and \(b_2(X,L)=h^2(X, \mathbb{C})\).NEWLINENEWLINEIn the paper under review, the author classifies polarized manifolds \((X,L)\) such that \(L\) is globally generated and \(b_2(X,L)=h^2(X, \mathbb{C})+1\). Concretely, for every integers \(i\) and \(j\) with \(0 \leq j \leq i \leq n\), we put NEWLINE\[NEWLINEC_j^i(X,L) := \sum_{l=0}^j (-1)^l \begin{pmatrix} n-i+l-1 \;\l \end{pmatrix} c_{j-l}(X)L^l.NEWLINE\]NEWLINE Then the \textit{\(i\)-th sectional Euler number} \(e_i(X,L)\) of \((X,L)\) is defined as NEWLINE\[NEWLINEe_i(X,L) := C_i^i(X,L) L^{n-i}NEWLINE\]NEWLINE and the \textit{\(i\)-th sectional Betti number} \(b_i(X,L)\) of \((X,L)\) is defined as NEWLINE\[NEWLINEb_i(X,L) := \begin{cases} e_0(X,L) & \roman{if} \;i=0 \\ (-1)^i( e_i(X,L) - \sum_{j=0}^{i-1}2(-1)^j h^i(X, \mathbb{C})) & \roman{if} \;1 \leq i \leq n. \end{cases} NEWLINE\]NEWLINE The main theorem in the paper under review is the following:NEWLINENEWLINETheorem 3.1. Let \((X, L)\) be a polarized manifold of dimension \(\geq 3\). Assume that \(L\) is globally generated. If \(b_2(X,L)=h^2(X, \mathbb{C})+1\), then \((X,L)\) is one of the following types.NEWLINENEWLINE\noindent \((a)\) \((\mathbb{Q}^n, \mathcal{O}_{\mathbb{Q}^n}(1))\).NEWLINENEWLINE\noindent \((b)\) \((\mathbb{P}^3, \mathcal{O}_{\mathbb{P}^3}(2))\).NEWLINENEWLINE\noindent \((c)\) A simple blowing up of \((X,L)\) of type \((b)\).NEWLINENEWLINE\noindent \((d)\) \((\mathbb{P}^1\times \mathbb{P}^1\times \mathbb{P}^1, \otimes_{i=1}^3 p_i^* \mathcal{O}_{\mathbb{P}^1}(1))\), where \(p_i\) is the \(i\)-th projection.NEWLINENEWLINE\noindent \((e)\) \((\mathbb{P}_S(\mathcal{E}), H(\mathcal{E}))\), where \(S\) is a smooth projective surface and \(\mathcal{E}\) is an ample and globally generated vector bundle of rank two on \(S\) with \(c_2(\mathcal{E})=2\). In particular, \((S,\mathcal{E})\) is one of the following.NEWLINENEWLINE\noindent (e.1) \((\mathbb{P}^2, \mathcal{O}_{\mathbb{P}^2}(1)\oplus\mathcal{O}_{\mathbb{P}^2}(2))\).NEWLINENEWLINE\noindent (e.2) \((\mathbb{Q}^2, \mathcal{O}_{\mathbb{Q}^2}(1)\oplus\mathcal{O}_{\mathbb{Q}^2}(1))\).NEWLINENEWLINE\noindent (e.3) \((\mathbb{P}_C(\mathcal{F}),\pi^*(\mathcal{G})\otimes H(\mathcal{F}))\), where \(C\) is an elliptic curve, \(\mathcal{F}\) and \(\mathcal{G}\) are indecomposable vector bundles of rank two on \(C\) with \(\deg \mathcal{F}=1\) and \(\deg \mathcal{G}=1\), and \(\pi: \mathbb{P}_C(\mathcal{F})\rightarrow C\) is the projection.NEWLINENEWLINE\noindent (e.4) \(S\) is a double covering of \(\mathbb{P}^2\), \(f:S\rightarrow \mathbb{P}^2\), and \(\mathcal{E}\simeq f^*(\mathcal{O}_{\mathbb{P}^2}(1))\oplus f^*(\mathcal{O}_{\mathbb{P}^2}(1))\).