Symplectic semi-characteristics

Abstract: We study the symplectic semi-characteristic of a 4n-dimensional closed symplectic manifold. First, we define the symplectic semi-characteristic using the mapping cone complex model of the primitive cohomology. Second, using a vector field with nondegenerate zero points, we prove a counting formula for the symplectic semi-characteristic. As corollaries, we obtain an Atiyah type vanishing property and the fact that the symplectic semi-characteristic is independent of the choices of symplectic forms.