发表机构
Utah State University; Northeastern University（犹他州立大学; 东北大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究将Polishchuk的泊松爆破准则应用于代数几何可积系统，证明仿射泊松概型沿可积系统纤维的爆破构成泊松概型族，并将结果推广到复半单李代数的伴随商，得到该族的平坦、锥形等性质及相关泊松几何结论。
AI 中文摘要
我们在代数几何可积系统的语境下运用了Polishchuk的泊松爆破准则。更具体地说，对于复数域上的每个仿射泊松概型$\frak{X}$，都可关联一个可积系统$\tau:\frak{X}\to\frak{B}$。我们证明了，$\frak{X}$沿$\tau$纤维的爆破是泊松概型，属于族$\tilde{\frak{X}\times\frak{B}}\to\frak{B}$，其中$\tilde{\frak{X}\times\frak{B}}$本身也是泊松概型。该结果随后被应用于有限维复半单李代数$\frak{g}$及其积分代数群$G$的伴随商$\tau:\frak{g}\to\frak{g}/\negthinspace/G=:\frak{c}$。我们证明族$\tilde{\frak{g}\times\frak{c}}\to\frak{c}$是平坦的、锥形的，且具有典范泊松哈密顿$G$-簇结构。我们还得到了该族纤维的泊松几何结果，这些纤维是$\frak{g}$沿正则伴随轨道闭包的爆破。
英文摘要
We leverage Polishchuk's Poisson blow-up criterion in the context of algebro-geometric integrable systems. In more detail, one may associate an integrable system $τ:\mathfrak{X}\longrightarrow\mathfrak{B}$ to each affine Poisson scheme $\mathfrak{X}$ over $\mathbb{C}$. We prove that the blow-ups of $\mathfrak{X}$ along fibers of $τ$ are Poisson schemes occurring in a family $\widetilde{\mathfrak{X}\times\mathfrak{B}}\longrightarrow\mathfrak{B}$, where $\widetilde{\mathfrak{X}\times\mathfrak{B}}$ is itself a Poisson scheme. This result is subsequently specialized to the adjoint quotient $τ:\mathfrak{g}\longrightarrow\mathfrak{g}/\!/G=:\mathfrak{c}$ of a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$ with integrating algebraic group $G$. We show that the family $\widetilde{\mathfrak{g}\times\mathfrak{c}}\longrightarrow\mathfrak{c}$ is flat, conical, and equipped with a canonical Poisson Hamiltonian $G$-variety structure. We also obtain Poisson-geometric results on the fibers of this family, which are blow-ups of $\mathfrak{g}$ along regular adjoint orbit closures.
