Symplectic semi-characteristics

TL;DR

The paper introduces the symplectic semi-characteristic $k(M,\omega)$ of a closed $4n$-dimensional symplectic manifold, defined via the primitive cohomology associated with the mapping cone model of filtered cohomology. It proves a parity counting formula: for a nondegenerate vector field $V$ on $M$, the mod $2$ value of $k(M,\omega)$ equals the number of zeros of $V$, established by constructing a skew-adjoint operator whose Atiyah–Singer mod $2$ index matches $k(M,\omega)$ and applying a symplectic version of Witten deformation with Bismut–Lebeau asymptotics. Consequences include an Atiyah-type vanishing property and the independence of $k(M,\omega)$ from the choice of symplectic form, with the 4n+2 case identified as open. The work blends Clifford actions, spectral analysis of a deformed Dirac-type operator, and localization techniques to connect topological invariants of primitive cohomology with geometric data of vector fields.

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.

Theorems & Definitions (42)

• Definition 1.3
• Definition 1.4
• Theorem 1.5
• Corollary 1.6
• Corollary 1.7
• Remark 1.8
• Proposition 2.1
• proof
• Lemma 2.2
• proof