Holomorphic Poisson deformations of Dolbeault algebras

Determine whether every holomorphic Poisson bivector on a complex manifold gives rise to a possibly $A_\infty$ non-commutative deformation of its Dolbeault algebra.

Background

The paper explains that infinitesimal deformations of the Dolbeault algebra of a compact complex manifold are controlled by Hochschild cohomology, whose components include holomorphic bivector fields in H0(X,∧2TX)H^0(X,\wedge^2T_X). Gerby and Kodaira–Spencer deformations have established geometric interpretations, but the corresponding non-commutative deformation theory for holomorphic Poisson bivectors is not known in general.

The authors note that the construction is available for a torus with a constant bivector, where a Fourier-mode star product produces a non-commutative Dolbeault algebra. The unresolved issue is whether an analogous construction exists for arbitrary holomorphic Poisson bivectors, potentially using an A∞A_\infty rather than strictly associative deformation.

References

While the formal deformation theories of $\mathcal A(X)$ and $X$ as a generalized complex manifold (as defined in ) match , it is not known whether any holomorphic poisson bivector does give rise to a (possibly $A_\infty$) non-commutative deformation of $\mathcal A(X)$.

— 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 discussing generalized deformations and the space $H^0(\wedge^2TX)$