On Fano 4-folds of index 1 and homogeneous vector bundles over Grassmannians
type: Item
instance of: scholarly article
title: On Fano 4-folds of index 1 and homogeneous vector bundles over Grassmannians
zbMATH Open document ID: 0826.14024
DOI: 10.1007/BF02571923
author: Oliver Küchle
published in: Mathematische Zeitschrift
publication date: 1995-11-27T00:00:00Z
full work available at URL: https://eudml.org/doc/174748
review text: Besides complete intersections of hypersurfaces in projective space, Grassmannians belong to the classical examples of Fano manifolds, by which we mean smooth algebraic varieties \(X\) defined over the field \(\mathbb{C}\) of complex numbers with ample anticanonical divisor \(- K_X\). The coindex of \(X\) is defined as \(\dim (X) - r + 1\) where \(r\), the index of \(X\), is the greatest integer which divides \(- K_X\) in \(\text{Pic} (X)\). -- Contrarily to the case of coindex \(\leq 3\), where the classification, due to Fujita, Iskovskih, Mori and Mukai, is complete up to a minor assumption, very little is known for larger coindex. We consider homogeneous vector bundles in the study of Fano manifolds of coindex 4: the Fano manifolds arising as zero schemes of sections in homogeneous vector bundles form a large class of examples and should be typical in some sense. We restrict ourselves to the case of bundles over Grassmannians of \(k\)-dimensional quotient spaces of \(\mathbb{C}^n\) which are homogeneous with respect to \(SL (n, \mathbb{C})\). The main result is the classification of all Fano 4-folds \(X\) of index 1 that are zero schemes of general global sections in direct sums of globally generated irreducible \(SL (n, \mathbb{C})\)-homogeneous vector bundles over Grassmannians. Moreover we compute the Hodge numbers \(h^{p,q} (X)\) as well as the Hilbert polynomials \(h^0 (X, {\mathcal O}_X (- tK_X))\) of these manifolds.
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 14J35
Mathematics Subject Classification ID: 14M15
zbMATH DE Number: 784089
zbMATH Keywords: Fano manifolds
zbMATH Keywords: coindex
zbMATH Keywords: bundles over Grassmannians
zbMATH Keywords: Hodge numbers
zbMATH Keywords: Hilbert polynomials
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 784089