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

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.