Advertisement

A note on the Hard Lefschetz property of symplectic structures

  • H. AndersenEmail author
Original Paper
  • 13 Downloads

Abstract

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.

Notes

References

  1. Audin, M.: Torus actions on symplectic manifolds (second revised edition). In: Progress in Mathematics, vol. 93. Birkhäuser, Basel (2004)CrossRefGoogle Scholar
  2. Cho, Y.: Hard Lefschetz property of symplectic structures on compact Kähler manifolds. Trans. Am. Math. Soc. 368(11), 8223–8248 (2016)CrossRefGoogle Scholar
  3. Draghici, T.: The Kähler cone versus the symplectic cone. Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie 42(90), 41–49 (1999). No. 1MathSciNetzbMATHGoogle Scholar
  4. Ibáñez, R., Rudyak, Y., Tralle, A., Ugarte, L.: On symplectically harmonic forms on six-dimensional nilmanifolds. Comment. Math. Helv. 76(1), 89–109 (2001)MathSciNetCrossRefGoogle Scholar
  5. Kirwan, F.C.: Cohomology of Quotients in Symplectic and Algebraic Geometry. Mathematical Notes, vol. 31. Princeton University Press, Princeton (1984)Google Scholar
  6. Lerman, E.: Symplectic cuts. Math. Res. Lett. 2(3), 247–258 (1995)MathSciNetCrossRefGoogle Scholar
  7. Li, H., Tolman, S.: Hamiltonian circle actions with minimal fixed sets. Int. J. Math. 23(8), 1250071 (2012)MathSciNetCrossRefGoogle Scholar
  8. McDuff, D., Salamon, D.: Introduction to Symplectic Topology. Oxford Mathematical Monographs. Oxford University Press, New York (1995)CrossRefGoogle Scholar
  9. McDuff, D., Tolman, S.: Topological properties of Hamiltonian circle actions. Int. Math. Res. Pap. 1(72826), 1–77 (2006)MathSciNetzbMATHGoogle Scholar
  10. Salamon, D.: Uniqueness of symplectic structures. Acta Math. Vietnam. 38(1), 123–144 (2013)MathSciNetCrossRefGoogle Scholar
  11. Schwald, M.: Fujiki relations and fibrations of irreducible symplectic varieties. E-print: arXiv:1701.09069v2 [math.AG]. October 18 (2017)
  12. Voisin, C.: Hodge structures on cohomology algebras and geometry. Math. Ann. 341(1), 39–69 (2008)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Managing Editors 2019

Authors and Affiliations

  1. 1.University of DallasIrvingUSA

Personalised recommendations