Autodual connection in the Fourier transform of a Higgs bundle
type: Item
instance of: scholarly article
title: Autodual connection in the Fourier transform of a Higgs bundle
zbMATH Open document ID: 1213.14066
author: Juhani Bonsdorff
published in: The Asian Journal of Mathematics
publication date: 2011-02-14T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ajm/1294789784
review text: A rank \(n\) and degree \(d\) Higgs bundle on a smooth projective complex curve \(X\), is a pair \(\mathsf{E}=(\mathcal E,\theta)\) where \(\mathcal E\) is a vector bundle on \(X\) of rank \(n\) and degree \(d\), and \(\theta:\mathcal E\to\mathcal E\otimes\Omega_X^1\). \(\mathsf{E}\) is semistable if \(\mu(\mathcal E')\leq\mu(\mathcal E)=\frac{d}{n}\), for every subbundle \(\mathcal E'\) of \(\mathcal E\) such that \(\theta(\mathcal E')\subset \mathcal E'\otimes\Omega_X^1\). \textit{N. J. Hitchin} [ Proc. Lond. Math. Soc., III. 55, 59--126 (1987; Zbl 0634.53045)] proved that the moduli space of semistable Higgs bundles over \(X\) contains the cotangent bundle of the moduli space of vector bundles on \(X\), and that it carries a natural hyper-Kähler structure, i.e. it is Kähler with respect to three complex structures \(I\), \(J\) and \(K\) satisfying \(IJK=-1\). Let \(\mathsf{E}=(\mathcal E,\theta)\) be a Higgs bundle of degree \(0\) on \(X\). Then, as proved by the author [J. Reine Angew. Math. 591, 21--48 (2006; Zbl 1094.14023)], its Fourier-Mukai transform \(\hat{\mathsf{E}}\) is an algebraic vector bundle on the cotangent bundle of the Jacobian of \(X\), \(T^*J(X)\cong J(X)\times H^0(X,\Omega_X^1)\), which is precisely the moduli space of rank \(1\) and degree \(0\) Higgs bundles on \(X\). The main result of this paper is the construction of a natural connection on \(\hat{\mathsf{E}}\) which is autodual with respect to the hyper-Kähler structure of \(T^*J(X)\). Given a complex manifold \(M\), its twistor space is \(\mathbf{Tw}(M)=M\times\mathbb{P}_{\mathbb{C}}^1\). Using a result of \textit{D. Kaledin} and \textit{M. Verbitsky} [Sel. Math., New Ser. 4, No. 2, 279--320 (1998; Zbl 0917.53006)] the construction of this autodual connection is reduced to the construction of a suitable holomorphic vector bundle on \(\mathbf{Tw}(T^*J(X))\), for which there is a complex analytic description (in arbitrary rank) worked out by \textit{C. Simpson} [in: Algebraic geometry. P…
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 53C07
Mathematics Subject Classification ID: 53C26
Mathematics Subject Classification ID: 53C28
Mathematics Subject Classification ID: 32C38
zbMATH DE Number: 5851255
zbMATH Keywords: Higgs bundles
zbMATH Keywords: Fourier transform
zbMATH Keywords: autodual connection, hyper-Kähler
zbMATH Keywords: \(\lambda\)-connections
zbMATH Keywords: twistor space
reviewed by: André G. Oliveira
MaRDI profile type: Publication
DOI: 10.4310/AJM.2010.V14.N2.A1
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description: scientific article; zbMATH DE number 5851255