Open Problems in Integrable Systems — Bi-Poisson vector spaces

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $V$ be a finite-dimensional complex vector space with alternating bilinear forms $A,B$, and let $J=\{A+\lambda B:\lambda\in\mathbb C\}\cup\{B\}$. Its generic corank $c$ is the minimum corank of these forms. Define \[\operatorname{Aut}(V,J)=\{T\in GL(V):A(Tu,Tv)=A(u,v),\ B(Tu,Tv)=B(u,v)\ \forall u,v\in V\}.\] Thus the automorphisms preserve each form, not merely the pencil as an unparameterized set. The Grassmannian $\operatorname{Gr}_k(V)$ consists of $k$-dimensional linear subspaces. A subspace $L$ is bi-Lagrangian if $A|_{L\times L}=B|_{L\times L}=0$ and $\dim L=(\dim V+c)/2$, the maximum possible common isotropic dimension. Let $LG(V,J)$ be the closed subvariety of the corresponding Grassmannian defined by these equations.

Statement recorded in the supplied source.

Give necessary and sufficient conditions on $(A,B)$ for $LG(V,J)$ to be a smooth algebraic variety, and classify its $\operatorname{Aut}(V,J)$-orbits. [S2]

Short English statement

Determine when the variety of maximal common isotropic subspaces is smooth and how pencil automorphisms partition it into orbits.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2100502 (AMR-020-0502). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 A. Bolsinov, V. S. Matveev, E. Miranda and S. Tabachnikov, Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018), arXiv:1804.03737, Problem 5.3 and bi-Lagrangian definition. https://arxiv.org/abs/1804.03737.