Classification of polarized manifolds by the second sectional Betti number
type: Item
instance of: scholarly article
title: Classification of polarized manifolds by the second sectional Betti number
zbMATH Open document ID: 1282.14012
author: Yoshiaki Fukuma
published in: Hokkaido Mathematical Journal
publication date: 2013-11-25T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.hokmj/1384273393
reviewed by: Hiroyuki Terakawa
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14C17
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14J35
Mathematics Subject Classification ID: 14J40
zbMATH DE Number: 6232366
zbMATH Keywords: polarized manifolds
zbMATH Keywords: classification
zbMATH Keywords: sectional invariants
zbMATH Keywords: sectional Euler numbers
zbMATH Keywords: sectional Betti numbers
zbMATH Keywords: sectional Hodge numbers
MaRDI profile type: Publication
OpenAlex ID: W2142511056
DOI: 10.14492/HOKMJ/1384273393
review text: 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 th…
description: scientific article; zbMATH DE number 6232366