English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Higher dimensional analogs of the Painlev\'e equations have been proposedfrom various aspects. In recent studies, 4-dimensional analogs of thePainlev\'e equations were classified into 40 types. The aim of the presentpaper is to geometrically characterize these 40 types of equations. For thispurpose, we study the autonomous limit of these equations and degeneration oftheir spectral curves. We obtain two functionally independent conservedquantities $H_1$ and $H_2$ for each system. We construct fibrations whose fiberat a general point $h_i $ is the spectral curve of the system with $H_i=h_i$for $i=1, 2$. The singular fibers at $H_{i}=\infty$ are one of the degeneratecurves of genus 2 classified by Namikawa and Ueno. Liu's algorithm enables usto give degeneration type of spectral curves for our 40 types of integrablesystems. This result is analogous to the following observation; spectral curvefibrations of the autonomous 2-dimensional Painlev\'e equations $P_{\rm I}$,$P_{\rm II}$, $P_{\rm IV}$, $P_{\rm III}^{D_8}$, $P_{\rm III}^{D_7}$, $P_{\rmIII}^{D_6}$, $P_{V}$ and $P_{\rm VI}$ are elliptic surfaces with the singularfiber at $H=\infty$ of Dynkin type  $E_8^{(1)}$, $E_7^{(1)}$, $E_6^{(1)}$, $D_8^{(1)}$, $D_7^{(1)}$, $D_6^{(1)}$,$D_5^{(1)}$ and $D_4^{(1)}$, respectively.

Autonomous limit of 4-dimensional Painlev\'e-type equations and degeneration of curves of genus two