Symplectic semi-characteristics

TL;DR

This paper introduces a new invariant called the symplectic semi-characteristic for 4n-dimensional closed symplectic manifolds, providing a counting formula and demonstrating its independence from symplectic form choices.

Contribution

It defines the symplectic semi-characteristic via primitive cohomology and proves a counting formula using vector fields with nondegenerate zeros, establishing key properties.

Findings

Derived a counting formula for the symplectic semi-characteristic.

Proved the invariance of the semi-characteristic under different symplectic forms.

Established an Atiyah type vanishing property.

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.