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CATEGORICAL PROOF OF HOLOMORPHIC ATIYAH–BOTT FORMULA

Part of: Algebraic geometry: Foundations General theory of categories and functors Homological algebra

Published online by Cambridge University Press:  20 December 2018

Grigory Kondyrev  and
Artem Prikhodko
Grigory Kondyrev
Affiliation:
National Research University Higher School of Economics, Russian Federation ([email protected])
Artem Prikhodko
Affiliation:
National Research University Higher School of Economics, Russian Federation, Center for Advanced Studies, Skoltech, Moscow, Russian Federation ([email protected])
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Abstract

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).

Keywords

tracesfixed-points formulas2-categories

MSC classification

Primary: 14A99: None of the above, but in this section 18A99: None of the above, but in this section
Secondary: 18G99: None of the above, but in this section
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 5 , September 2020 , pp. 1739 - 1763
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Copyright
© Cambridge University Press 2018

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References

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