Examples of Compact Lefschetz Solvmanifolds
2002; Publication Committee for the Tokyo Journal of Mathematics; Volume: 25; Issue: 2 Linguagem: Inglês
10.3836/tjm/1244208853
ISSN0387-3870
Autores Tópico(s)Algebraic Geometry and Number Theory
ResumoIntroduction.Let (M 2m , ω) be a compact symplectic manifold.A symplectic manifold (M, ω) is called a Lefschetz manifold if the mapping ∧ω m-1 :is an isomorphism.We also say that (M, ω) has the Hard Lefschetz property, if the mapping ∧ω k :By a solvmanifold we mean a homogeneous space G/Γ , where G is a simply-connected solvable Lie group and Γ is a lattice, that is, a discrete co-compact subgroup of G.A solvable Lie algebra g is called completely solvable if ad(X) : g → g has only real eigenvalues for each X ∈ g.Benson and Gordon [BG1] have proved that no non-toral compact nilmanifolds are Lefschetz manifolds for any symplectic structure to show that a non-toral compact nilmanifold does not admit any Kähler structure.Moreover, they conjecture the following :Let G be a simply-connected completely solvable Lie group and Γ a lattice of G. Then a compact solvmanifold G/Γ admits a Kähler structure if and only if it is a torus.
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