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Approximating classifying spaces by smooth projective varieties · Around

2009

Approximating classifying spaces by smooth projective varieties

Ekedahl, Torsten

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We prove that for every reductive algebraic group $H$ with centre of positive dimension and every integer $K$ there is a smooth and projective variety $X$ and an algebraic $H$-torsor $P \to X$ such that the classifying map $X \to \Bclass H$ induces an isomorphism in cohomology in degrees $\le K$.

This is then applied to show that if $G$ is a connected non-special group there is a $G$-torsor $P \to X$ for which we do not have $[P]=[G][X]$ in the (completion of the) Grothendieck ring of varieties.

Built on

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J.-P. Serre, Sur la topologie des variétés algébriques en caractéristique p p , Symposium internacional de topología algebraica, Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, pp. 24–53
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M. F. Atiyah and F. Hirzebruch, Analytic cycles on complex manifolds , Topology 1
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J. Milnor, Morse theory , Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963
1963
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P. Deligne, Theorie de Hodge, III , Publ. Math. IHES (1974), no. 44, 5–78
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P. Deligne, P. Griffiths, J. Morgan, and D. Sullivan, Real homotopy theory of Kähler manifolds , Invent. Math. 29
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Cited alongside, same era.

Then

F. C. Kirwan, Cohomology of quotients in symplectic and algebraic geometry , Mathematical notes, vol. 31, Princeton University Press, Princeton, NJ, 1984
1984
Later among the works it cites.
T. Ekedahl, Two examples of smooth projective varieties with nonzero Massey products , Algebra, algebraic topology and their interactions (Stockholm, 1983), Lecture Notes in Math., vol. 1183, Springer, Berlin, 1986, pp. 128–132
1986
Later among the works it cites.
T. Ekedahl, The Grothendieck group of algebraic stacks , 2009, arXiv:math/0903.3143
2009
Closest in time.

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