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CATEGORICAL PROOF OF HOLOMORPHIC ATIYAH–BOTT FORMULA

  • Grigory Kondyrev (a1) and Artem Prikhodko (a2)

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).

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References

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1.Atiyah, M. F. and Bott, R, A Lefschetz fixed point formula for elliptic complexes I, Ann. of Math. (2) 86(2) (1967), 374407.
2.Atiyah, M. F. and Bott, R., A Lefschetz fixed point formula for elliptic complexes II. Applications, Ann. of Math. (2) 88(3) (1968), 451491.
3.Ben-Zvi, D., Francis, J. and Nadler, D., Integral transforms and Drinfeld centers in derived algebraic geometry, J. Amer. Math. Soc. 23(4) (2010), 909966.
4.Ben-Zvi, D. and Nadler, D., Nonlinear traces, Preprint, 2013, available at arXiv:1305.7175v3.
5.Gaitsgory, D. and Rozenblyum, N., A study in derived algebraic geometry, available at http://www.math.harvard.edu/∼gaitsgde/GL/.
6.Lefschetz, S., Intersections and transformations of complexes and manifolds, Trans. Amer. Math. Soc. 28 (1926), 149.
7.Lurie, J., Higher algebra, available at http://www.math.harvard.edu/∼lurie/papers/HA.pdf.
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CATEGORICAL PROOF OF HOLOMORPHIC ATIYAH–BOTT FORMULA

  • Grigory Kondyrev (a1) and Artem Prikhodko (a2)

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