On the sectional geometric genus of quasi-polarized varieties. II
type: Item
instance of: scholarly article
title: On the sectional geometric genus of quasi-polarized varieties. II
zbMATH Open document ID: 1080.14013
author: Yoshiaki Fukuma
published in: Manuscripta Mathematica
publication date: 2004-07-01T00:00:00Z
review text: 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.
reviewed by: Vincenzo di Gennaro
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14J32
Mathematics Subject Classification ID: 14J35
Mathematics Subject Classification ID: 14J40
Mathematics Subject Classification ID: 14J45
zbMATH DE Number: 2078009
zbMATH Keywords: nef and big line bundle
zbMATH Keywords: abelian variety
zbMATH Keywords: Calabi-Yau variety
zbMATH Keywords: Fano variety
zbMATH Keywords: Castelnuovo-Harris bound
zbMATH Keywords: normal variety
zbMATH Keywords: Gorenstein variety
zbMATH Keywords: Cohen-Macaulay variety
MaRDI profile type: Publication
full work available at URL: s00229-003-0434-9
OpenAlex ID: W2048569044
DOI: 10.1007/S00229-003-0434-9
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description: scientific article; zbMATH DE number 2078009