Behrend function

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

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

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

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

is the degree of the virtual fundamental class of X.

References

• K. Behrend, Donaldson-Thomas invariants via microlocal geometry, arXiv math 0507523.