The LLV Algebra for Primitive Symplectic Varieties with Isolated Singularities

TL;DR

This paper extends the algebraic understanding of the Lie algebra associated with the intersection cohomology of primitive symplectic varieties with isolated singularities, providing new proofs and insights without relying on hyperk"ahler metrics.

Contribution

It establishes an explicit isomorphism of the total Lie algebra with a special orthogonal algebra, offering a new algebraic proof for properties of irreducible holomorphic symplectic manifolds.

Findings

The Lie algebra for intersection cohomology is isomorphic to a special orthogonal algebra.

Provides an algebraic proof for properties of irreducible holomorphic symplectic manifolds.

Studies the structure of intersection cohomology as a Lie algebra representation.

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.