Fréchet–Fredholm properties for more general non-commutative manifolds

Establish that the curved differential graded algebras associated with non-commutative manifolds beyond non-commutative tori are Fréchet–Fredholm and possess the functional-analytic properties required for the paper’s Grauert theory.

Background

The paper develops its Grauert theorem for Fréchet–Fredholm curved differential graded algebras. The principal examples treated are Dolbeault algebras of compact manifolds and generalized or non-commutative tori, for which elliptic estimates and approximation properties yield the required Fredholm framework.

The authors indicate that analogous properties are only conjectural for more general non-commutative manifolds. Establishing those properties would extend the developed categorical and Fourier–Mukai machinery beyond the torus setting.

References

One key observation in this paper is that the curved dg-algebras $\mathcal A(X,0,B)$ (for general compact $X$) and $\mathcal A(V/\Gamma,\beta,B)$ (for tori and conjecturally for more general non-commutative manifolds) come equipped with very favorable functional analytic properties, making them into Fréchet-Fredholm cdg-algebras ('c' standing for curved), Definition \ref{DefFreFred}.

— A Non Commutative Grauert Theorem and Fourier Mukai Duality for Generalized Complex Tori  (2610.01945 - Antweiler et al., 1 Oct 2026) in Section 0, Introduction, paragraph titled “Contents of the paper”