A correspondence between the moduli spaces of vector bundles over a curve
type: Item
instance of: scholarly article
title: A correspondence between the moduli spaces of vector bundles over a curve
zbMATH Open document ID: 0946.14014
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2000-06-29T00:00:00Z
review text: The author establishes a correspondence between a smooth moduli space of vector bundles over a curve and a self-product of the Jacobian, and proves the nontriviality of the Griffiths group of the moduli space for a general curve. Let \(X\) be a smooth algebraic curve over \(\mathbb{C}\), and \(r\) and \(d\) be two coprime positive integers. Let \(\mathcal N\) be the moduli space of isomorphism classes of stable rank-\(r\) vector bundles over \(X\) with a fixed determinant of degree \(d\). Then, it is well-known that \(\mathcal N\) is smooth and there exists a universal bundle \(\mathcal E\) over \(X \times \mathcal N\). The characteristic classes \(a_k \in H^{2k}(X \times \mathcal N, \mathbb{Q})\) (\(2 \leq k \leq r\)) of \(\mathbb{P}(\mathcal E)\) give rise to homomorphisms from \(H_1(X, \mathbb{Q})\) to \(H^{2k-1}(\mathcal N, \mathbb{Q})\) by the slant product operation. These homomorphisms induce an algebra homomorphism from \(H_*(J^{r-1}, \mathbb{Q})\) to \(H^*(\mathcal N, \mathbb{Q})\), where \(J^{r-1}\) is the \((r-1)\)-fold product of the Jacobian \(J\). The main result of the paper says that the above algebra homomorphism is induced by an algebraic cycle on the product \(\mathcal N \times J^{r-1}\). Moreover, this algebraic cycle is canonical as an element of the Chow ring (cycles modulo rational equivalence) of \(\mathcal N \times J^{r-1}\). As an application of the main result, the author constructs non-zero elements in the Griffiths group of the moduli space \(\mathcal N\) for a general curve \(X\). These elements are obtained from the non-zero elements of the Jacobian \(J\) of a general curve discovered by G. Ceresa.
reviewed by: Zhenbo Qin
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 14C15
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 14M12
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14C25
zbMATH DE Number: 1347786
zbMATH Keywords: moduli space
zbMATH Keywords: vector bundles
zbMATH Keywords: algebraic cycles
zbMATH Keywords: Griffith group
zbMATH Keywords: product of the Jacobian
zbMATH Keywords: Chow ring
author: Indranil Biswas
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178224809
OpenAlex ID: W2013739730
cites work: Q4364029
cites work: Canonical generators of the cohomology of moduli parabolic bundles on curves
cites work: On the mapping class group action on the cohomology of the representation space of a surface
cites work: A remark on Hodge cycles on moduli spaces of rank 2 bundles over curves.
cites work: Algebraic cohomology of the moduli space of rank 2 vector bundles on a curve
cites work: C is not algebraically equivalent to \(C^-\) in its Jacobian
cites work: Q3803252
cites work: Q3328651
cites work: Torelli Groups and Geometry of Moduli Spaces of Curves
cites work: Completions of mapping class groups and the cycle $C - C^-$
cites work: Remarks on Classical Invariant Theory
cites work: On the cohomology ring of the moduli space of rank 2 vector bundles on a curve
cites work: Characteristic Classes of Stable Bundles of Rank 2 Over an Algebraic Curve
DOI: 10.2748/TMJ/1178224809
description: scientific article; zbMATH DE number 1347786