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Moduli of Abelian Varieties, Vinberg θ-Groups, and Free Resolutions

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Commutative Algebra

Abstract

We present a systematic approach to studying the geometric aspects of Vinberg θ-representations. The main idea is to use the Borel-Weil construction for representations of reductive groups as sections of homogeneous bundles on homogeneous spaces, and then to study degeneracy loci of these vector bundles. Our main technical tool is to use free resolutions as an “enhanced” version of degeneracy loci formulas. We illustrate our approach on several examples and show how they are connected to moduli spaces of Abelian varieties. To make the article accessible to both algebraists and geometers, we also include background material on free resolutions and representation theory.

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Acknowledgements

We thank Manjul Bhargava, Igor Dolgachev, Wei Ho, Steven Kleiman, Bjorn Poonen, and Jack Thorne for helpful discussions and Damiano Testa for explaining the proof of Theorem 1. We also thank Igor Dolgachev for pointing out numerous references related to this work. Finally, we thank an anonymous referee for making some suggestions.

We also thank Federico Galetto and Witold Kraśkiewicz for assistance with some computer calculations. The software LiE [39] and Macaulay2 27] were helpful in our work.

Steven Sam was supported by an NDSEG fellowship while this work was done. Jerzy Weyman was partially supported by NSF grant DMS-0901185.

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Authors and Affiliations

  1. Université de Versailles Saint-Quentin-en-Yvelines, Versailles, France

    Laurent Gruson

  2. University of California, Berkeley, CA, USA

    Steven V. Sam

  3. Northeastern University, Boston, MA, USA

    Jerzy Weyman

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  1. Laurent Gruson
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Correspondence to Jerzy Weyman .

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  1. Cornell University, 547 Malott Hall, Ithaca, 14853, New York, USA

    Irena Peeva

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Gruson, L., Sam, S.V., Weyman, J. (2013). Moduli of Abelian Varieties, Vinberg θ-Groups, and Free Resolutions. In: Peeva, I. (eds) Commutative Algebra. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5292-8_13

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