Behrend function

In algebraic geometry, a 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

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

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 technical difficulties (e.g., what is the Chow group of a stack?), the definition extends to moduli stacks such as the moduli stack of semi-stable vector bundles or that of stable maps.

References

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