Atiyah–Bott formula
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Short description: On the cohomology ring of the moduli stack of principal bundles
In algebraic geometry, the Atiyah–Bott formula says[1] the cohomology ring
- [math]\displaystyle{ \operatorname{H}^*(\operatorname{Bun}_G(X), \mathbb{Q}_l) }[/math]
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 [math]\displaystyle{ \operatorname{Bun}_G(X) }[/math].
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. doi:10.1098/rsta.1983.0017. Bibcode: 1983RSPTA.308..523A.
- Gaitsgory, Dennis; Lurie, Jacob (2019), Weil's Conjecture for Function Fields, Annals of Mathematics Studies, 199, Princeton, NJ: Princeton University Press, http://www.math.harvard.edu/~lurie/papers/tamagawa.pdf
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