Behrend function

{{Short description|Function in algebraic geometry}}

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

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

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

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

is the degree of the virtual fundamental class

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

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).