Behrend Function

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

[math]\displaystyle{ \nu_X: X \to \mathbb{Z} }[/math]

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

[math]\displaystyle{ \chi(X, \nu_X) = \sum_{n \in \mathbb{Z}} n \, \chi(\{\nu_X = n\}) }[/math]

is the degree of the virtual fundamental class

[math]\displaystyle{ [X]^{\text{vir}} }[/math]

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

• "Donaldson–Thomas type invariants via microlocal geometry", Annals of Mathematics, 2nd Ser. 170 (3): 1307–1338, 2009, doi:10.4007/annals.2009.170.1307 .

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