论文标题
谎言代数和叶子的模量空间
Moduli spaces of Lie algebras and foliations
论文作者
论文摘要
令$ x $成为复数上平滑的投影变化，$ s（d）$ $ h^0（x，x，\ nathcal {t} x）$的参数化$ d $ d $ d $二维谎言subergebras。本文致力于研究Moduli空间$ \ text {Inv} $在$ x $上的$ x $的几何{Inv} $，$ \ MATHCAL {f} \ in \ text {Inv} $由Lie组动作引起。对于每一个$ \ mathfrak {g} \在s（d）$中，一个人都可以考虑相应的元素$ \ mathcal {f}（\ mathfrak {g}）\ in \ text {inv} $，其通用叶子与$ \ expfrak of $ \ expfrak的动作相吻合。我们表明，在轻度假设下进行分层$ \ coprod_i s（d）_i \ to s（d）$，此任务产生同构$ ϕ：\ coprod_i s（d）_i \ to \ tock \ text {Inv} $ \ mathcal {f}（\ mathfrak {g}）$。这为文献中独立出现的许多结果提供了共同的解释。我们还建立了由谎言群体行动引起的新稳定叶子家庭。
Let $X$ be a smooth projective variety over the complex numbers and $S(d)$ the scheme parametrizing $d$-dimensional Lie subalgebras of $H^0(X,\mathcal{T} X)$. This article is dedicated to the study of the geometry of the moduli space $\text{Inv}$ of involutive distributions on $X$ around the points $\mathcal{F}\in \text{Inv}$ which are induced by Lie group actions. For every $\mathfrak{g} \in S(d)$ one can consider the corresponding element $\mathcal{F}(\mathfrak{g})\in \text{Inv}$, whose generic leaf coincides with an orbit of the action of $\exp(\mathfrak{g})$ on $X$. We show that under mild hypotheses, after taking a stratification $\coprod_i S(d)_i\to S(d)$ this assignment yields an isomorphism $ϕ:\coprod_i S(d)_i\to \text{Inv}$ locally around $\mathfrak{g}$ and $\mathcal{F}(\mathfrak{g})$. This gives a common explanation for many results appearing independently in the literature. We also construct new stable families of foliations which are induced by Lie group actions.
