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CATEGORICAL PROOF OF HOLOMORPHIC ATIYAH–BOTT FORMULA
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 19 / Issue 5 / September 2020
- Published online by Cambridge University Press:
- 20 December 2018, pp. 1739-1763
- Print publication:
- September 2020
-
- Article
- Export citation
-
Given a
$2$-commutative diagram
in a symmetric monoidal
$(\infty ,2)$-category 
$\mathscr{E}$ where 
$X,Y\in \mathscr{E}$ are dualizable objects and 
$\unicode[STIX]{x1D711}$ admits a right adjoint we construct a natural morphism 
$\mathsf{Tr}_{\mathscr{E}}(F_{X})\xrightarrow[{}]{~~~~~}\mathsf{Tr}_{\mathscr{E}}(F_{Y})$ between the traces of 
$F_{X}$ and 
$F_{Y}$, respectively. We then apply this formalism to the case when 
$\mathscr{E}$ is the 
$(\infty ,2)$-category of 
$k$-linear presentable categories which in combination of various calculations in the setting of derived algebraic geometry gives a categorical proof of the classical Atiyah–Bott formula (also known as the Holomorphic Lefschetz fixed point formula).
