Gómez, Tomás L.; Sols, Ignacio Stable Higgs \(G\)-sheaves. (English) Zbl 1155.14010 Rev. Mat. Iberoam. 24, No. 2, 703-719 (2008). A Higgs bundle over a projective scheme \(X\) is a vector bundle \(E\), together with a homomorphism \(\theta: E\to E\otimes\Omega_X\), such that \(\theta\wedge\theta=0\). The authors extend this notion of Higgs bundles to principal \(G\)-sheaves on a smooth projective complex scheme \(X\) of any dimension, where \(G\) is an arbitrary reductive structure group. The starting point is a principal \(G\)-sheaf \(P\) over a torsion free sheaf \(E\). The authors define a \(G\)-Higgs field on \(P\) by means of a homomorphism \(\Theta:E\oplus(\zeta \otimes \mathcal O_X)\), where \(\zeta\) is the center. Then the authors define the wedge product \(\Theta\wedge\Theta\) by extending sections arising from a natural exact sequence. Higgs \(G\)-sheaves are \(G\)-sheaves for which \(\Theta\wedge\Theta=0\). The notion of semistability for Higgs \(G\)-fields is introduced by means of compatibility with orthogonal algebra filtrations on \(E\). The authors prove the existence of a quasi-projective coarse moduli space of classes of semistable Higgs \(G\)-sheaves, with fixed numerical invariants. Reviewer: Luca Chiantini (Siena) Cited in 1 Document MSC: 14D22 Fine and coarse moduli spaces 14D20 Algebraic moduli problems, moduli of vector bundles Keywords:Higgs bundles × Cite Format Result Cite Review PDF Full Text: DOI Euclid EuDML References: [1] Anchouche, B. and Biswas, I.: Einstein-Hermitian connections on polystable principal bundles over a compact Kähler manifold. Amer. J. Math. 123 (2001), 207-228. · Zbl 1007.53026 · doi:10.1353/ajm.2001.0007 [2] Balaji, V.: Principal bundles on projective varieties and the Donaldson-Uhlenbeck compactification. J. Differential Geom. 76 (2007), no. 3, 351-398. · Zbl 1121.14037 [3] Beauville, A., Narasimhan, M. S. and Ramanan, S.: Spectral curves and the generalised theta divisor. J. Reine Angew. Math. 398 (1989), 169-179. · Zbl 0666.14015 · doi:10.1515/crll.1989.398.169 [4] Gómez, T., Langer, A., Schmitt, A. and Sols, I.: Moduli spaces for principal bundles in large characteristic. To appear in Proceedings “International Workshop on Teichmüller Theory and Moduli Problems”, Allahabad 2006. India . [5] Gómez, T. and Sols, I.: Moduli space of principal sheaves over projective varieties. Annals of Math. (2) 161 (2005) no. 2, 1037-1092. · Zbl 1079.14018 · doi:10.4007/annals.2005.161.1037 [6] Hitchin, N. J.: The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3) 55 (1987), 59-126. · Zbl 0634.53045 · doi:10.1112/plms/s3-55.1.59 [7] Huybrechts, D. and Lehn, M.: The geometry of moduli spaces of sheaves . Aspects of Mathematics E31 . Friedr. Vieweg & Sohn, Braunschweig, 1997. · Zbl 0872.14002 [8] Schmitt, A. H. W.: Projective moduli for Hitchin pairs. Internat. J. Math. 9 (1998), no. 1, 107-118. Erratum: 11 (2000), no. 4, 589. · Zbl 0936.14009 · doi:10.1142/S0129167X98000075 [9] Schmitt, A. H. W.: Singular principal bundles over higher-dimensional manifolds and their moduli spaces. Int. Math. Res. Not. 2002 , no. 23, 1183-1209. · Zbl 1034.14017 · doi:10.1155/S1073792802107069 [10] Schmitt, A. H. W.: A universal construction for moduli spaces of decorated vector bundles over curves. Transform. Groups 9 (2004), no. 2, 167-209. · Zbl 1092.14042 · doi:10.1007/s00031-004-7010-6 [11] Schmitt, A. H. W.: A closer look at semistability for singular principal bundles. Int. Math. Res. Not. 2004 , no. 62, 3327-3366. · Zbl 1093.14507 · doi:10.1155/S1073792804132984 [12] Schmitt, A. H. W.: A relative version of geometric invariant theory. In preparation (2007). [13] Simpson, C.: Higgs bundles and local systems. Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5-95. · Zbl 0814.32003 · doi:10.1007/BF02699491 [14] Simpson, C.: Moduli of representations of the fundamental group of a smooth projective variety I. Inst. Hautes Études Sci. Publ. Math. 79 (1994), 47-129. · Zbl 0891.14005 · doi:10.1007/BF02698887 [15] Simpson, C.: Moduli of representations of the fundamental group of a smooth projective variety II. Inst. Hautes Études Sci. Publ. Math. 80 (1995), 5-79. · Zbl 0891.14006 · doi:10.1007/BF02698895 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.