Amazing examples of nonrational smooth spectral surfaces

TL;DR

This paper constructs new smooth projective surfaces of general type with specific divisor properties, linking algebraic geometry to the structure of commutative subalgebras of differential operators, revealing unique spectral surfaces without isospectral deformations.

Contribution

It provides the first explicit examples of such surfaces satisfying particular divisor conditions related to differential operator algebras.

Findings

Constructed smooth projective surfaces with specified divisor properties.

Linked geometric conditions to algebraic structures of differential operators.

Showed these spectral surfaces do not admit isospectral deformations.

Abstract

In this paper we construct first examples of smooth projective surfaces of general type satisfying the following conditions: there are 1) an ample integral curve $C$ with $C^2=1$ and $h^0(X,O_X(C))=1$; \quad 2) a divisor $D$ with $(D, C)_X=g(C)-1$, $h^i(X,O_X(D))=0$, $i=0,1,2$, and $h^0(X, O_X(D+C))=1$. Such conditions arise from necessary and sufficient conditions for the existence of non-trivial commutative subalgebras of rank one in $\hat{D}$, a completion of the algebra of partial differential operators in two variables, which can be thought of as a simple algebraic analogue of the algebra of analytic pseudodifferential operators on a manifold. We extract these conditions by elaborating the classification theorem of commutative subalgebras in $\hat{D}$ due to the second author for the case of rank one subalgebras. Amazingly, the commutative subalgebras with such spectral surfaces do not admit isospectral deformations.