Behrend function

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

\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

\chi (X,\nu _{X})=\sum _{n\in \mathbb {Z} }n\,\chi (\{\nu _{X}=n\})

[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, MR 2600874, arXiv:math/0507523 , doi:10.4007/annals.2009.170.1307 .

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