Symplectic Lefschetz fibrations on adjoint orbits

Abstract

We prove that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We describe the topology of the regular and singular fibres, in particular we calculate their middle Betti numbers.

Deformations of Noncompact Calabi-Yau threefolds

“…Corollary. [GGSM1,Cor. 4.5] The homology of a regular fibre coincides with the homology of F Θ \ W · H Θ .…”

“…Corollary. [GGSM1,Cor. 5.1] The homology of the singular fibre though wH Θ , w ∈ W, coincides with that of…”

Compactifications of adjoint orbits and their Hodge diamonds

“…The compact subgroup K of G cuts out the subadjoint orbit Ad (K ) H 0 , which can be identified with the flag manifold F H 0 = G/P H 0 where P H 0 is the parabolic subgroup associated to H 0 . In [GGSM1] the symplectic structure on Ad (G) H 0 is chosen as the imaginary part of the Hermitian form inherited from g. With this choice,…”

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A recent theorem of [GGSM1] showed that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We investigate the behaviour of their fibrewise compactifications.

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“…[ GGSM1] showed the existence of the structure of SLFs on adjoint orbits of semisimple Lie algebras. These adjoint orbits are not compact.…”