论文标题
赫森伯格品种和泊松片
Hessenberg varieties and Poisson slices
论文作者
论文摘要
这项工作追求了涉及黑森伯格品种，泊松几何形状和精美压实的谎言理论思想。更详细地，可以将符合性的哈密顿式$ g $ -variety $μ：g \ times \ mathcal {s} \ longrightarrow \ mathfrak {g} $与每个复杂的semisimple lie algebra $ \ mathfrak $ \ mathfrak {G} $ \ mathcal {s} \ subseteq \ mathfrak {g} $。这个品种是比拉夫斯基的Hyperkähler片之一，它对于Moore和Tachikawa在拓扑量子场理论方面的工作至关重要。它还与两个log Simpletic Hamiltonian $ g $ -Varieties $ \Overlineμ_ {\ Mathcal {s}}}：\ Overline {G \ Times \ Times \ Mathcal {S}} $ν：\ mathrm {hess} \ longrightarrow \ mathfrak {g} $。前者是在奇妙的紧凑型$ \ overline {g} $的日志cotangent束中的泊松横向，而后者是赫森伯格品种的标准家族。 $ \Overlineμ$和$ν$的每一个都是$μ$的纤维压实。 我们利用泊松切片理论来关联上述纤维的压缩。我们的主要结果是一个规范$ g $ equivariant bimeromormormormormormormormormormormormormormormormormormormormorm {hess} \ cong \ cong \ overline {g \ times \ times \ mathcal {s}} $，$ \ mathfrak {g} $。这种双态形态被证明是综合中的哈密顿$ g $ g $变异同构，并且与bălibanu获得的泊松同构兼容。如果$ \ mathfrak {g} = \ mathfrak {sl} _2 $，我们还表明我们的双形态形态是一种生物形态，并且我们猜想是任意$ \ mathfrak {g} $的情况。我们通过讨论我们猜想对赫森伯格品种的含义的结论。
This work pursues a circle of Lie-theoretic ideas involving Hessenberg varieties, Poisson geometry, and wonderful compactifications. In more detail, one may associate a symplectic Hamiltonian $G$-variety $μ:G\times\mathcal{S}\longrightarrow\mathfrak{g}$ to each complex semisimple Lie algebra $\mathfrak{g}$ with adjoint group $G$ and fixed Kostant section $\mathcal{S}\subseteq\mathfrak{g}$. This variety is one of Bielawski's hyperkähler slices, and it is central to Moore and Tachikawa's work on topological quantum field theories. It also bears a close relation to two log symplectic Hamiltonian $G$-varieties $\overlineμ_{\mathcal{S}}:\overline{G\times\mathcal{S}}\longrightarrow\mathfrak{g}$ and $ν:\mathrm{Hess}\longrightarrow\mathfrak{g}$. The former is a Poisson transversal in the log cotangent bundle of the wonderful compactification $\overline{G}$, while the latter is the standard family of Hessenberg varieties. Each of $\overlineμ$ and $ν$ is known to be a fibrewise compactification of $μ$. We exploit the theory of Poisson slices to relate the fibrewise compactifications mentioned above. Our main result is a canonical $G$-equivariant bimeromorphism $\mathrm{Hess}\cong\overline{G\times\mathcal{S}}$ of varieties over $\mathfrak{g}$. This bimeromorphism is shown to be a Hamiltonian $G$-variety isomorphism in codimension one, and to be compatible with a Poisson isomorphism obtained by Bălibanu. We also show our bimeromorphism to be a biholomorphism if $\mathfrak{g}=\mathfrak{sl}_2$, and we conjecture that this is the case for arbitrary $\mathfrak{g}$. We conclude by discussing the implications of our conjecture for Hessenberg varieties.
