A note on the Hard Lefschetz property of symplectic structures

Abstract

In (Trans Am Math Soc 368(11), 8223–8248, 2016), Cho has constructed the first known example of a deformation equivalence between a Kahler form and a symplectic non-Hard Lefschetz form. He achieved this in complex dimension 3 by using a principal $$S^{1}$$-bundle P on a K3-surface N and taking a symplectic cut along extremal level sets of a particular Hamiltonian moment map on the tube $$P\times [-1,1]$$. In this note, we explore the analogous construction when N is a hyperKahler Hilbert scheme $$K3^{[2]}$$ of two points on a K3-surface. We take a brief excursion into elementary $$S^{1}$$-equivariant cohomology in order to study the Hard Lefschetz cone of closed simple Hamiltonian-$$S^{1}$$ manifolds of real dimension 10 with vanishing odd cohomology, and we thereupon obtain a criterion that determines when those with diffeomorphic fixed components (of the circle action) of dimension 8 are of Hard Lefschetz type. Using this criterion, we find another deformation equivalence between a Kahler form and a symplectic non-Hard Lefschetz form, thereby successfully extending Cho’s result to the next dimension.