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Behrend function

Function in algebraic geometry

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Source updated April 24, 2024

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

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

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).

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