2022/11/13 by Benjamin Tighe
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebra over a field #Algebraic structures and combinatorial models #Gravitational singularity #Mathematical analysis #Mathematics #Nonlinear Waves and Solitons #Pure mathematics #Symplectic geometry #math.AG
paper · pdf · doi:10.46298/epiga.2025.12186
published as Épijournal de Géométrie Algébrique, Volume 9 (March 28, 2025) epiga:12186 · 41 pages; Final journal version; new subsection on LLV algebra for symplectic orbifolds
arxiv created 2025/03/26 · openalex publication_date 2025/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/05/21 · arxiv updated 2026/08/05
We extend results of Looijenga--Lunts and Verbitsky and show that the total Lie algebra \mathfrak g for the intersection cohomology of a primitive symplectic variety X with isolated singularities is isomorphic to \mathfrak g ≅ \mathfrakso((IH2(X, \mathbb Q), QX)⊕ \mathfrak h), where QX is the intersection Beauville--Bogomolov--Fujiki form and \mathfrak h is a hyperbolic plane. This gives a new, algebraic proof for irreducible holomorphic symplectic manifolds which does not rely on the hyperk"ahler metric. Along the way, we study the structure of IH^*(X, \mathbb Q) as a \mathfrakg-representation -- with particular emphasis on the Verbitsky component, multidimensional Kuga--Satake constructions, and Mumford--Tate algebras -- and give some immediate applications concerning the P = W conjecture for primitive symplectic varieties. Comment: 41 pages; Final journal version; new subsection on LLV algebra for symplectic orbifolds