论文标题
具有孤立奇点的原始符号变种的LLV代数
The LLV Algebra for Primitive Symplectic Varieties with Isolated Singularities
论文作者
论文摘要
我们扩展了looijenga-lunts和verbitsky的结果，并表明，对于原始的符号品种$ x $与孤立奇异性的相交共同体的总体代数$ \ mathfrak g $是$ \ mathfrak g \ mathfrak g \ cong \ congfrak {s o s o s s o s o n Math（ q_x \ right）\ oplus \ mathfrak h \ right），$ q_x $是beauville-bogomolov--fujiki形式和$ \ mathfrak h $是一架夸张的平面。这为不依赖于HyperKähler指标的不可还原性全身形态流形提供了新的代数证明。 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.
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 hyperkähler 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.
