Degeneracy of triality-symmetric morphisms

Anderson, Dave

doi:10.2140/ant.2012.6.689

ANT

2012

DOI: 10.2140/ant.2012.6.689

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Degeneracy of triality-symmetric morphisms

Dave Anderson

Abstract: We define a new symmetry for morphisms of vector bundles, called triality symmetry, and compute Chern class formulas for the degeneracy loci of such morphisms. In an appendix, we show how to canonically associate an octonion algebra bundle to any rank-2 vector bundle.

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Cited by 3 publications

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References 20 publications

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“…This is the situation considered in [Ha-Tu], for example, with F = E * and symmetric or skew-symmetric maps. We investigate the G 2 analogue of this problem in [An2].…”

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confidence: 99%

Chern class formulas for $G_{2}$ Schubert loci

Anderson¹

2011

Trans. Amer. Math. Soc.

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Abstract. We define degeneracy loci for vector bundles with structure group G 2 and give formulas for their cohomology (or Chow) classes in terms of the Chern classes of the bundles involved. When the base is a point, such formulas are part of the theory for rational homogeneous spaces developed by BernsteinGelfand-Gelfand and Demazure. This has been extended to the setting of general algebraic geometry by Giambelli-Thom-Porteous, Kempf-Laksov, and Fulton in classical types; the present work carries out the analogous program in type G 2 . We include explicit descriptions of the G 2 flag variety and its Schubert varieties, and several computations, including one that answers a question of W. Graham.In appendices, we collect some facts from representation theory and compute the Chow rings of quadric bundles, correcting an error in a paper by Edidin and Graham.

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“…This is the situation considered in [Ha-Tu], for example, with F = E * and symmetric or skew-symmetric maps. We investigate the G 2 analogue of this problem in [An2].…”

mentioning

confidence: 99%

Chern class formulas for $G_{2}$ Schubert loci

Anderson¹

2011

Trans. Amer. Math. Soc.

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Lie algebra and Dynkin index

Luan

2023

Communications in Algebra

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A construction of Shatz strata in the polystable $ G_2 $-bundles moduli space using Hecke curves

Antón-Sancho

2024

era

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<p>Let $ X $ be a compact Riemann surface of genus $ g\geq 2 $ and $ M(G_2) $ be the moduli space of polystable principal $ G_2 $-bundles over $ X $. The Harder-Narasimhan types of the bundles induced a stratification of the moduli space $ M(G_2) $ called Shatz stratification. In this paper, a description of the Shatz strata of the unstable locus of $ M(G_2) $ corresponding to certain family of Harder-Narasimhan types (specifically, those of the form $ (\lambda, \mu, 0, -\mu, -\lambda) $ with $ \mu < \lambda\leq 0 $) was given. For this purpose, a family of vector bundles was constructed in which a 3-form and a 2-form were defined so that it was proved that they were strictly polystable principal $ G_2 $-bundles. From this, it was proved that, when the genus of $ X $ was $ g\geq 12 $, these Shatz strata were the disjoint union of a family of $ G_2 $-Hecke curves in $ M(G_2) $ that will be constructed along the paper. Therefore, the presented results provided an advance in the knowledge of the geometry of $ M(G_2) $ through the study of its Shatz strata and presented a methodological innovation, by using Hecke curves for this study.</p>

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Degeneracy of triality-symmetric morphisms

Abstract: We define a new symmetry for morphisms of vector bundles, called triality symmetry, and compute Chern class formulas for the degeneracy loci of such morphisms. In an appendix, we show how to canonically associate an octonion algebra bundle to any rank-2 vector bundle.

Cited by 3 publications

References 20 publications

Chern class formulas for $G_{2}$ Schubert loci

Chern class formulas for $G_{2}$ Schubert loci

Lie algebra and Dynkin index

A construction of Shatz strata in the polystable $ G_2 $-bundles moduli space using Hecke curves

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