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In `F33f`_`[algebraic geometry`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebraic_geometry]`_`f, the `!Behrend function`! of a scheme `*X`*, introduced by `F33f`_`[Kai Behrend`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Kai_Behrend]`_`f, is a constructible function

ν ν X : X → → Z {\\displaystyle \\nu _{X}:X\\to \\mathbb {Z} }

such that if `*X`* is a quasi-projective proper moduli scheme carrying a `F33f`_`[symmetric obstruction theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Symmetric_obstruction_theory]`_`f, then the `!weighted Euler characteristic`!

χ χ ( X , ν ν X ) = ∑ ∑ n ∈ ∈ Z n χ χ ( { ν ν X = n } ) {\\displaystyle \\chi (X,\\nu _{X})=\\sum _{n\\in \\mathbb {Z} }n\\,\\chi (\\{\\nu _{X}=n\\})}

is the degree of the `F33f`_`[virtual fundamental class`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Virtual_fundamental_class]`_`f

[ X ] vir {\\displaystyle [X]^{\\text{vir}}}

of `*X`*, which is an element of the zeroth `F33f`_`[Chow group`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Chow_group]`_`f of `*X`*. Modulo some solvable technical difficulties (e.g., what is the `F33f`_`[Chow group of a stack`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Chow_group_of_a_stack]`_`f?), the definition extends to moduli stacks such as the moduli stack of stable sheaves (the `F33f`_`[Donaldson–Thomas theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Donaldson–Thomas_theory]`_`f) or that of `F33f`_`[stable maps`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Stable_map]`_`f (the `F33f`_`[Gromov–Witten theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Gromov–Witten_theory]`_`f).

>>References

• `:citerefbehrend2009`a`F33f`_`[Behrend, Kai`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Kai_Behrend]`_`f (2009), "Donaldson–Thomas type invariants via microlocal geometry", `*`F33f`_`[Annals of Mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Annals_of_Mathematics]`_`f`*, 2nd Ser., `!170`! (3): 1307–1338, `F33f`_`[arXiv`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ArXiv_(identifier)]`_`f:math/0507523, `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.4007/annals.2009.170.1307, `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 2600874.

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