A vanishing property about the 1-filtered cohomology groups of (4n+2)-dimensional closed symplectic manifolds

TL;DR

The paper addresses whether the even-degree part of the $1$-filtered cohomology of $(4n+2)$-dimensional closed symplectic manifolds has even dimension. It defines the $1$-filtered cochain complex with $\psi=\omega\wedge\omega$, introduces the parity invariant $\ell(M,\psi)$, and proves $\ell(M,\psi)=0$ using a dimension-adjusted skew-adjoint operator and a Witten deformation. The main contributions include the construction of the skew-adjoint operator in the $4n+2$ setting, a mod $2$-index argument, and several explicit examples that illustrate the vanishing and contrast with the primitive case in $4n$ dimensions. This establishes a Vanishing/Parity property for $1$-filtered cohomology that parallels Euler-characteristic-type results in odd dimensions, enriching the theory of filtered symplectic invariants and their geometric implications.

Abstract

This note is a follow-up to our previous work arXiv:2505.14496. For any (4n+2)-dimensional closed symplectic manifold, we find that the dimension of the even-degree part of its 1-filtered cohomology is even, similar to the vanishing property of the classical Euler characteristic of an odd-dimensional closed manifold. We prove our result by constructing and then deforming a skew-adjoint operator. This process follows the methods in arXiv:2505.14496 but needs adjustments on signs and the power of the symplectic form.

Theorems & Definitions (12)

• Theorem 1.2
• Remark 1.3
• Proposition 2.1
• Proposition 2.2
• Remark 2.3
• Corollary 2.4
• Proposition 2.5
• proof
• Example 3.1
• Example 3.2