截断多项式与超曲面李代数的泊松几何
Polynomial degeneration and the Poisson geometry of truncated polynomials
AI总结：
本文研究了超曲面李代数的辛几何，通过分析截断多项式与组合关系，提出了一种变形辛形式的方法，生成新的泊松结构，并展示了偶维情况下自然的泊松结构。
摘要：
我们研究超曲面李代数上的辛形式。这些是广泛推广的$b^k$-泊松结构的研究对象，由Miranda、Scott及其合作者广泛研究，其几何与截断多项式在组合下的群密切相关。它们诱导出通常为辛的泊松结构，并在codimension 1子流形W上降秩。然而，与$b^k$-泊松结构不同，沿W的辛分层可以有非零的辛变化，反映超曲面李代数阶数消失的障碍。除了研究这些李代数的辛几何外，本文还对李代数de Rham复形进行了详细研究，并开发了一种沿k- jets特征流形中路径变形辛形式的方法。结果使我们能够产生大量新的泊松结构实例。最后，我们构造了通用的超曲面李代数，并证明在偶维情况下它们具有自然的泊松结构。
英文摘要：
We develop a formalism for studying geometric structures that degenerate to polynomial order along a hypersurface $W \subset M$. We then demonstrate it in the study of Poisson geometry, where it leads to methods for constructing generically symplectic Poisson structures with non-trivial symplectic variation along their degeneracy locus. This is in contrast to $b/\log$-symplectic and $b^k$-symplectic structures, where this variation always vanishes. Our main insight is that the higher residue data along the hypersurface is controlled by a group of transverse diffeomorphisms, which in our case is the group $G_k$ of degree-$k$ truncated polynomials. We show that the symplectic variation of our Poisson structures is determined by the obstruction to lifting a $G_k$-representation of the fundamental group $π_1(W)$ to $G_{k+1}$, and we construct maps from a $G_k$-character variety into the moduli space of Poisson structures, with the variation detecting the non-triviality of the resulting families.
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