Behrend function

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

${\nu }_X:X\to \mathbb{\mathbb{Z}}$

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

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

is the degree of the virtual fundamental class

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

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.