On a Commutative Ring of Two Variable Differential Operators with Matrix Coefficients

TL;DR

This paper constructs a commutative ring of two-variable matrix differential operators linked to a ring of meromorphic functions on a rational manifold derived from complex projective spaces.

Contribution

It introduces a novel construction of commutative rings of matrix differential operators associated with a specific rational manifold.

Findings

Establishment of an isomorphism between the differential operator ring and meromorphic functions

Explicit construction of the commutative ring with matrix coefficients

Connection between differential operators and algebraic geometry of rational manifolds

Abstract

In this work, we construct commutative rings of two variable matrix differential operators that are isomorphic to a ring of meromorphic functions on a rational manifold obtained from the $CP^1\times CP^1$ by identification of two lines with the pole on a certain rational curve.