Compactifications of adjoint orbits and their Hodge diamonds

Abstract

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. Expressing adjoint orbits and fibres as affine varieties in their Lie algebra, we compactify them to projective varieties via homogenisation of the defining ideals. We find that their Hodge diamonds vary wildly according to the choice of homogenisation, and that extensions of the potential to the compactification must acquire degenerate singularities. E. G. was supported by a Simmons Associateship Grant from ICTP, Italy. LEFSCHETZ FIBRATIONS ON ADJOINT ORBITSLet H 0 be an element in the Cartan subalgebra of a semisimple Lie algebra g, let O (H 0 ) denote its adjoint orbit and 〈·, ·〉 the Cartan-Killing form. It is proved in [GGSM1] that for each regular element H ∈ g, the function

Birationally equivalent Landau–Ginzburg models on cotangent bundles and adjoint orbits

“…We consider here the minimal case (as in Definition 2.6) which we denote by LG n . These Landau-Ginzburg models were shown to satisfy the conjecture of Katzarkov, Kontsevich and Pantev about three new Hodge theoretical invariants that take into consideration the potential, see [8,18] and also [5] for some more features about Hodge diamonds of compactifications.…”

Some Landau–Ginzburg models viewed as rational maps

“…Remark 1.1. Observe that the algebraic geometric method will in general produce singular compactifications, see [BCG,Sec. 6], whereas that the Lie theoretical method always embeds the orbit into a product of smooth flag manifolds.…”

“…We may then obtain a compactification by homogenizing the ideal I . In general the resulting compactification will be very singular, see [BCG,Sec. 6].…”