Behrend function

Function in algebraic geometry

In algebraic geometry, the Behrend function of a scheme X, introduced by Kai Behrend, is a constructible function

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

such that if X is a quasi-projective proper moduli scheme carrying a symmetric obstruction theory, then the weighted Euler characteristic

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

is the degree of the virtual fundamental class

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

of X, which is an element of the zeroth Chow group of X. Modulo some solvable technical difficulties (e.g., what is the Chow group of a stack?), the definition extends to moduli stacks such as the moduli stack of stable sheaves (the Donaldson–Thomas theory) or that of stable maps (the Gromov–Witten theory).

References

  • Behrend, Kai (2009), "Donaldson–Thomas type invariants via microlocal geometry", Annals of Mathematics, 2nd Ser., 170 (3): 1307–1338, arXiv:math/0507523, doi:10.4007/annals.2009.170.1307, MR 2600874.


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