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# A categorification of the quantum Lefschetz principle

Abstract : The quantum Lefschetz formula explains how virtual fundamental classes (or structure sheaves) of moduli stacks of stable maps behave when passing from an ambient target scheme to the zero locus of a section. It is only valid under special assumptions (genus $0$, regularity of the section and convexity of the bundle). In this paper, we give a general statement at the geometric level removing these assumptions, using derived geometry. Through a study of the structure sheaves of derived zero loci we deduce a categorification of the formula in the $\infty$-categories of quasi-coherent sheaves. We also prove that Manolache's virtual pullbacks can be constructed as derived pullbacks, and use them to get back the classical quantum Lefschetz formula when the hypotheses are satisfied.
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Preprints, Working Papers, ...
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Cited literature [40 references]

https://hal.archives-ouvertes.fr/hal-02476725
Contributor : David Kern Connect in order to contact the contributor
Submitted on : Wednesday, February 12, 2020 - 8:36:42 PM
Last modification on : Wednesday, November 3, 2021 - 9:18:39 AM

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qLefschetzCateg.pdf
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• HAL Id : hal-02476725, version 1

### Citation

David Kern. A categorification of the quantum Lefschetz principle. 2020. ⟨hal-02476725⟩

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