Abstract
We introduce the Néron–Severi Lie algebra of a Soergel module and we determine it for a large class of Schubert varieties. This is achieved by investigating which Soergel modules admit a tensor decomposition. We also use the Néron–Severi Lie algebra to provide an easy proof of the well-known fact that a Schubert variety is rationally smooth if and only if its Betti numbers satisfy Poincaré duality.
Highlights
Let X be a smooth complex projective variety of dimension n and ρ ∈ H2(X, R) be the Chern class of an ample line bundle on X
In [LL] Looijenga and Lunts defined the Neron–Severi Lie algebra gNS(X) of X to be the Lie algebra generated by all the gρ with ρ an ample class
Looijenga and Lunts’ initial motivation was to find a “universal” primitive decomposition of H(X), not depending on any choice: this is achieved by considering the decomposition of H(X) into irreducible gNS(X)-modules, which always exists as one can prove that gNS(X) is semisimple
Summary
Let X be a smooth complex projective variety of dimension n and ρ ∈ H2(X, R) be the Chern class of an ample line bundle on X. If W is finite, since gNS(w) is semisimple and Bw is indecomposable as R-module ( as gNS(w)-module), it follows that Bw is an irreducible gNS(w)module Looijenga and Lunts went on to compute gNS(X) for a flag variety X = G/B They prove that it is “as big as possible,” meaning that it is the complete Lie algebra of endomorphisms of H(X) preserving a non-degenerate (either symmetric or antisymmetric depending on the parity of dim X) bilinear form on H(X). For the vast majority of Schubert varieties the Neron–Severi Lie algebra is “as big as possible.”
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