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Algebraic rational cells and equivariant intersection theory

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Abstract

We provide a notion of algebraic rational cell with applications to intersection theory on singular varieties with torus action. Based on this notion, we study \(\mathbb {Q}\)-filtrable varieties: algebraic varieties where a torus acts with isolated fixed points, such that the associated Białynicki-Birula decomposition consists of algebraic rational cells. We show that the rational equivariant Chow group of any \(\mathbb {Q}\)-filtrable variety is freely generated by the classes of the cell closures. We apply this result to group embeddings, and more generally to spherical varieties.

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Acknowledgments

The research in this paper was done during my visits to the Institute des Hautes Études Scientifiques (IHES) and the Max-Planck-Institut für Mathematik (MPIM). I am deeply grateful to both institutions for their support, outstanding hospitality, and excellent working conditions. A very special thank you goes to Michel Brion for the productive meeting we had at IHES, from which this paper received much inspiration. I would also like to thank the support that I received, as a postdoctoral fellow, from Sabancı Üniversitesi, the Scientific and Technological Research Council of Turkey (TÜBİTAK), and the German Research Foundation (DFG). I am further grateful to the referees for very helpful comments and suggestions that improved the clarity of the article.

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Correspondence to Richard P. Gonzales.

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Supported by the Institut des Hautes Études Scientifiques, the Max-Planck-Institut für Mathematik, TÜBİTAK Project No. 112T233, and DFG Research Grant PE2165/1-1.

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Gonzales, R.P. Algebraic rational cells and equivariant intersection theory. Math. Z. 282, 79–97 (2016). https://doi.org/10.1007/s00209-015-1533-5

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  • DOI: https://doi.org/10.1007/s00209-015-1533-5

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