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Atiyah–Bott formula

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On the cohomology ring of the moduli stack of principal bundles For other uses, see Atiyah–Bott fixed-point theorem.

In algebraic geometry, the Atiyah–Bott formula says the cohomology ring

\operatorname {H} ^{*}(\operatorname {Bun} _{G}(X),\mathbb {Q} _{l})

of the moduli stack of principal bundles is a free graded-commutative algebra on certain homogeneous generators. The original work of Michael Atiyah and Raoul Bott concerned the integral cohomology ring of $\operatorname {Bun} _{G}(X)$ .

See also

Borel's theorem, which says that the cohomology ring of a classifying stack is a polynomial ring.

Notes

Gaitsgory & Lurie 2019, § 6.2.

References

Atiyah, Michael F.; Bott, Raoul (1983). "The Yang-Mills equations over Riemann surfaces". Philosophical Transactions of the Royal Society of London. Ser. A. 308 (1505): 523–615. Bibcode:1983RSPTA.308..523A. doi:10.1098/rsta.1983.0017. MR 0702806. S2CID 13601126.
Gaitsgory, Dennis; Lurie, Jacob (2019), Weil's Conjecture for Function Fields (PDF), Annals of Mathematics Studies, vol. 199, Princeton, NJ: Princeton University Press, MR 3887650

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