algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, the Behrend function of a scheme ''X'', introduced by

Kai Behrend Kai Behrend is a German mathematician. He is a professor at the University of British Columbia in Vancouver, British Columbia, Canada. His work is in algebraic geometry and he has made important contributions in the theory of algebraic stacks, ...

, is a

constructible function In complexity theory, a time-constructible function is a function ''f'' from natural numbers to natural numbers with the property that ''f''(''n'') can be constructed from ''n'' by a Turing machine in the time of order ''f''(''n''). The purpose of ...

\nu_X: X \to \mathbb

such that if ''X'' is a quasi-projective proper moduli scheme carrying a

symmetric obstruction theory In algebraic geometry, given a Deligne–Mumford stack ''X'', a perfect obstruction theory for ''X'' consists of: # a perfect two-term complex E = ^ \to E^0/math> in the derived category D(\text(X)_) of quasi-coherent étale sheaves on ''X'', an ...

, then the weighted Euler characteristic :

\chi(X, \nu_X) = \sum_ n \, \chi(\)

is the degree of the

virtual fundamental class In mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. ...

of ''X'', which is an element of the zeroth

Chow group In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties ...

of ''X''. Modulo some solvable technical difficulties (e.g., what is the

Chow group of a stack In algebraic geometry, the Chow group of a stack is a generalization of the Chow group of a variety or scheme to stacks. For a quotient stack X = /G/math>, the Chow group of ''X'' is the same as the ''G''-equivariant Chow group of ''Y''. A key d ...

?), the definition extends to moduli stacks such as the moduli stack of stable sheaves (the

Donaldson–Thomas theory In mathematics, specifically algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of ...

) or that of

stable map In mathematics, specifically in symplectic topology and algebraic geometry, one can construct the moduli space of stable maps, satisfying specified conditions, from Riemann surfaces into a given symplectic manifold. This moduli space is the es ...

s (the Gromov–Witten theory).

References

*. Geometry {{algebraic-geometry-stub