Coordinate independence of the sym-characteristic variety

Determine whether the sym-characteristic variety defined from the asymptotic non-invertibility directions of the coordinate-dependent total symbol of a matrix differential operator of infinite order is independent of the choice of local coordinates.

Background

For scalar differential operators of infinite order, the paper introduces the sym-characteristic variety by examining the asymptotic directions in which the total symbol fails to be invertible. Unlike the abstract characteristic variety defined using holomorphic microlocal operators, this construction depends explicitly on a chosen coordinate system and is therefore potentially non-intrinsic.

The paper notes that the scalar construction is useful and that the abstract characteristic variety is contained in the sym-characteristic variety, but leaves unresolved whether the latter is invariant under changes of coordinates in the matrix-valued setting. Establishing such invariance would clarify whether this more concrete symbol-based object is geometrically well defined.

References

As the total symbol is coordinate dependent, it is not clear whether the definition of $A$ is, in general, independent on the choice of coordinates.

— Supersymmetry, differential operators of infinite order and theta functions  (2608.12846 - Kapranov, 13 Aug 2026) in Section 2, paragraph “Characteristic varieties via zeroes of the total symbol”