The LLV Algebra for Primitive Symplectic Varieties with Isolated Singularities

Abstract

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 $$\mathfrak g \cong \mathfrak{so}\left(\left(IH^2(X, \mathbb Q), Q_X\right)\oplus \mathfrak h\right),$$ where $Q_X$ 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 $\mathfrak{g}$-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.