Topological Euler number (awaiting review)

The (topological) Euler number of a smooth projective variety $X$ over a field of characteristic zero is the quantity $$ \chi = \sum_{i=0}^{2\dim X} (-1)^i \cdot H^i(X(\mathbb{C}), \mathbb{Z}) $$ where the cohomology is singular cohomology on the topological space $X(\mathbb{C})$.