[Paper Review] On some questions related to the Krichever correspondence

Denis Vasilievich Osipov, A. B. Zheglov|arXiv (Cornell University)|Oct 11, 2006

Algebraic structures and combinatorial models13 references4 citations

TL;DR

This paper advances the two-dimensional Krichever correspondence by constructing explicit examples on $\mathbb{P}^2$, deriving new KP-type equations from generalized KP-hierarchies on two-dimensional local skew fields, and classifying these skew fields via invariants. It establishes a link between algebraic geometry and integrable systems through the Krichever map in higher dimensions, showing that nontrivial automorphisms lead to solvable hierarchies and deriving a new nontrivial KP equation with parameter $a$ that reduces to the classical KP equation when $a=0$. The key contribution is a generalized KP-hierarchy with noncommutative structure and a new integrable equation incorporating $a$-dependent terms.

ABSTRACT

We investigate various new properties and examples of one-dimensional and two-dimensional Krichever correspondence developed by Parshin. In particular, we give explicit examples of the Krichever-Parshin map for various plane curves, we introduce analogs of the Schur pairs in a two-dimensional local field and show that they are oft geometrical. At the end we investigate analogs of the KP hierarchy for two-dimensional local skew-fields with arbitrary commutation law instead of the usual law of Weyl algebra. We derive for these hierarchies new partial differential equations, which coincide with the usual KP equation for certain values of parameters.

Motivation & Objective

• To clarify the two-dimensional Krichever correspondence through explicit geometric examples on $\mathbb{P}^2$.
• To generalize the one-dimensional Krichever correspondence to higher dimensions using two-dimensional local fields.
• To derive new KP-type equations from generalized KP-hierarchies on two-dimensional local skew fields.
• To classify two-dimensional local skew fields of characteristic 0 using invariants $ (n, \xi, i_n, r_n, c, a_n) $.
• To investigate the role of canonical automorphisms in the solvability of the KP-hierarchy and identify nontrivial cases.

Proposed method

• Constructs the Krichever map for surfaces using data $ (X, C, p, \mathcal{F}, e_p, t, u) $, where $ X $ is a surface, $ C $ a curve, $ p $ a point, and $ \mathcal{F} $ a vector bundle.
• Uses two-dimensional local fields $ k((t))((u)) $ via completion at a point $ p $ on a curve $ C $, with $ u $ a local equation of $ C $ and $ t $ a local parameter.
• Applies Lemma 1 to compute the Krichever subspace $ W \subset k((t))((u))^{\oplus r} $ as $ k[y_1, \dots, y_l]((f)) $, where $ f $ defines the curve away from $ p $.
• Derives a generalized KP-hierarchy in Lax form: $ \partial_n L / \partial t_n = [(L^n)_+, L] $, with $ L \in z^{-1} + K_- \otimes k[[\dots, t_m, \dots]] $.
• Analyzes the hierarchy under the assumption $ \alpha = \text{id} $, focusing on the case $ i=1 $, $ r=0 $, $ c=1 $, leaving $ a $ as the only nontrivial parameter.
• Eliminates auxiliary variables to derive a new nontrivial KP equation: $ (4u_t - u''' - 12u u')' = 3u_{yy} + 6a(2x^{-2}u'' - x^{-2}u_y - x^{-1}u''' + x^{-1}u'_y - 2x^{-3}u') $.

Experimental results

Research questions

• RQ1 How can the two-dimensional Krichever correspondence be explicitly realized on $ \mathbb{P}^2 $ for singular or tangent curves?
• RQ2 What are the conditions under which the generalized KP-hierarchy on two-dimensional local skew fields yields nontrivial integrable equations?
• RQ3 How do the invariants $ (n, \xi, i_n, r_n, c, a_n) $ classify two-dimensional local skew fields of characteristic 0?
• RQ4 What is the role of the canonical automorphism $ \alpha $ in the solvability of the KP-hierarchy?
• RQ5 How does the parameter $ a $ modify the classical KP equation, and what is the structure of the resulting nontrivial equation?

Key findings

• For a quadric curve $ C $ on $ \mathbb{P}^2 $ with tangent line $ D $ at $ p $, the Krichever subspace is $ W = k[\alpha]((\alpha^2 u / t^2)) $, where $ \alpha = y/z $, and $ u = (y/x)^2 + (z/x)^2 - 2z/x $.
• When $ D \cdot C = 2p $, the Krichever subspace is explicitly computed as $ k[\alpha]((\alpha^2 u / t^2)) $, confirming Lemma 1 in a geometric setting.
• The generalized KP-hierarchy is trivial if the canonical automorphism $ \alpha $ is nontrivial, but becomes nontrivial only when $ \alpha = \text{id} $ and $ i=1 $, $ r=0 $, $ c=1 $, leaving $ a $ as the key parameter.
• The derived KP equation is $ (4u_t - u''' - 12u u')' = 3u_{yy} + 6a(2x^{-2}u'' - x^{-2}u_y - x^{-1}u''' + x^{-1}u'_y - 2x^{-3}u') $, which reduces to the classical KP equation when $ a=0 $.
• The skew field $ K $ is isomorphic to a finite-dimensional extension of $ k((u))((z)) $ with $ z u z^{-1} = \xi u + u^{\delta'_{i_n}} z^{i_n} + \cdots $, and classification depends on $ (n, \xi, i_n, r_n, c, a_n) $.
• Every such skew field admits a decomposition $ K = K_+ \oplus K_- $, where $ K_- $ consists of elements with negative order, enabling the Lax formulation of the KP-hierarchy.