Ample and spanned vector bundles of minimal curve genus
type: Item
instance of: scholarly article
title: Ample and spanned vector bundles of minimal curve genus
zbMATH Open document ID: 0853.14022
DOI: 10.1007/BF01273345
author: H. Maeda
author: Antonio Lanteri
author: Andrew J. Sommese
published in: Archiv der Mathematik
publication date: 1997-01-06T00:00:00Z
review text: Let \({\mathcal E}\) be an ample and spanned vector bundle of rank \(n - 1\) on a compact complex manifold \(X\) of dimension \(n \geq 3\) and let \(g(X, {\mathcal E})\) be the curve genus, defined by \[ 2g (X, {\mathcal E}) - 2 = (K_X + c_1 ({\mathcal E})) c_{n - 1} ({\mathcal E}). \] The fact that \({\mathcal E}\) is ample and spanned implies the inequality \(g(X, {\mathcal E}) \geq h^{1,0} (X)\). The main result of the paper is the characterization of pairs \((X, {\mathcal E})\) as above satisfying the condition \[ g(X, {\mathcal E}) = h^{1,0} (X)\tag{*} \] A consequence of this is that (*) implies the vanishing of \(h^0 (m(K_X + \text{det} {\mathcal E}))\) for all integers \(m \geq 1\). The conjecture related to the equivalence between (*) and the vanishing of \(h^0 (K_X + \text{det} {\mathcal E})\), suggested by well known adjunction theoretic facts, is also discussed and proved in some special instances.
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14J40
Mathematics Subject Classification ID: 14F05
zbMATH DE Number: 937059
zbMATH Keywords: adjunction theory
zbMATH Keywords: Albanese variety
zbMATH Keywords: ample and spanned vector bundle
zbMATH Keywords: curve genus
reviewed by: Antonio Lanteri
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 937059