A note on the Hard Lefschetz property of symplectic structures

  • H. AndersenEmail author
Original Paper


In (Trans Am Math Soc 368(11), 8223–8248, 2016), Cho has constructed the first known example of a deformation equivalence between a Kähler 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 hyperKähler 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 Kähler form and a symplectic non-Hard Lefschetz form, thereby successfully extending Cho’s result to the next dimension.



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Authors and Affiliations

  1. 1.University of DallasIrvingUSA

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