The Hodge conjecture for certain moduli varieties
type: Item
instance of: scholarly article
title: The Hodge conjecture for certain moduli varieties
zbMATH Open document ID: 0868.14005
DOI: 10.1007/BF02836872
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1996-09-24T00:00:00Z
review text: Let \(C\) be a smooth irreducible curve of genus \(g\geq 2\) over the complex numbers, and let \(M(n,\xi)\) be the moduli space of stable vector bundles \(V\) of rank \(n\) on \(C\) such that \(\text{det }V\cong \xi\), where \(\xi\) is a line bundle of degree \(d\) such that \(n\) and \(d\) are coprime. In this paper, the author proves the Hodge conjecture Hodge\((p,p)\) for the cases \(n=3\), \(g=2\) and \(n=2\), \(g=3,4\). In the case \(n=2\), results of \textit{M. Thaddeus} [Invent. Math. 117, No. 2, 317-353 (1994)] show that \(M(2,\xi)\) is dominated by a variety that is obtained from a projective space by blowing up and down along smooth centers of dimension at most 3, and the Hodge conjecture follows by an argument of \textit{J. P. Murre} [Nederl. Akad. Wet., Proc., Ser. A 80, 230-232 (1977; Zbl 0352.14006)]. The paper is mainly devoted to the case \(n=3\). The author uses the method of normal functions (due to H. Poincaré and S. Lefschetz, and further developed by P. Griffiths and S. Zucker). In this case the normal functions are not associated to a Lefschetz pencil, but to a fibration over a curve whose fibers are projective bundles over \(M(2,\xi)\). The proof is finished using Hecke correspondences.
reviewed by: J. Nagel
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14D07
zbMATH DE Number: 903372
zbMATH Keywords: moduli space
zbMATH Keywords: Hodge conjecture
zbMATH Keywords: normal functions
zbMATH Keywords: Hecke correspondences
Wikidata QID: Q123114421
author: V. Balaji
MaRDI profile type: Publication
cites work: Intermediate Jacobian of Some Moduli Spaces of Vector Bundles on Curves
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cites work: Periods of integrals on algebraic manifolds. III: Some global differential-geometric properties of the period mapping
cites work: Q4122017
cites work: Stable bundles of rank 2 and odd degree over a curve of genus 2
cites work: Stable and unitary vector bundles on a compact Riemann surface
cites work: Moduli of pairs and generalized theta divisors
cites work: Q3043286
cites work: Special divisors and vector bundles
cites work: Stable pairs, linear systems and the Verlinde formula
cites work: Q4116549
cites work: Q3677856
cites work: Generalized intermediate Jacobians and the theorem on normal functions
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description: scientific article; zbMATH DE number 903372