Behrend function

Last updated March 13, 2019

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

Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.

Kai Behrend is a German mathematician. He is a professor at the University of British Columbia in Vancouver, British Columbia, Canada.

\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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References

Behrend, Kai (2009), "Donaldson–Thomas type invariants via microlocal geometry", Annals of Mathematics , 2nd Ser., 170 (3): 1307–1338, arXiv: math/0507523  , doi:10.4007/annals.2009.170.1307, MR 2600874 .

The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study.

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