Coisotropic deformations of algebraic varieties and integrable systems

TL;DR

This paper explores coisotropic deformations of algebraic varieties, revealing their connection to integrable systems like dKP and WDVV equations, with a focus on elliptic curves and Poisson structures.

Contribution

It introduces the concept of coisotropic deformations of algebraic varieties and links them to well-known integrable hydrodynamical systems, especially for elliptic curves.

Findings

Coisotropic deformations are governed by integrable equations like dKP and WDVV.

Deformations of elliptic curves are analyzed in detail.

The choice of Poisson structure is crucial for these deformations.

Abstract

Coisotropic deformations of algebraic varieties are defined as those for which an ideal of the deformed variety is a Poisson ideal. It is shown that coisotropic deformations of sets of intersection points of plane quadrics, cubics and space algebraic curves are governed, in particular, by the dKP, WDVV, dVN, d2DTL equations and other integrable hydrodynamical type systems. Particular attention is paid to the study of two- and three-dimensional deformations of elliptic curves. Problem of an appropriate choice of Poisson structure is discussed.