Abstract: The Gauss-Bonnet-Chern theorem states that for a smooth compact complex manifold X, the integration of the top Chern class of X over X is the topological Euler characteristic of X. In order to study Chern class for singular varieties or schemes, R. MacPherson introduced the notion of local Euler obstruction of singular varieties. The local Euler obstruction is an integer value constructible function on X, and the constant function 1_X can be written down as the linear combination of local Euler obstructions. A characteristic class for a local Euler obstruction was defined by using Nash blow-ups, and is called the Chern-Mather class or Chern-Schwartz-MacPherson class. The Chern-Schwartz-MacPherson class of the constant function 1_X is defined as the Chern class for X.
Inspired by gauge theory in higher dimension and string theory, the curve counting theory via stable coherent sheaves was constructed by Donaldson-Thomas on projective 3-folds, which is now called the Donaldson-Thomas theory. In the case of the Calabi-Yau threefolds, the Donaldson-Thomas invariants are proved by K. Behrend to be "weighted Euler characteristic" of the moduli space X, where the weights come from the local Euler obstruction of the moduli space X. In this talk I will survey some results of the Donaldson-Thomas invariants along this line, and talk about one case that how the Behrend weighted Euler characteristic is related the Y. Kiem and J. Li's cosection localization invariants.