Quantisation of derived Poisson structures

We prove that every $0$-shifted Poisson structure on a derived Artin $n$-stack admits a curved $A_{\mathrm{\infty }}$ deformation quantisation whenever the stack has perfect cotangent complex; in particular, this applies to LCI schemes, where it gives a DQ algebroid quantisation. Whereas the Kontsevich–Tamarkin approach to quantisation for smooth varieties hinges on invariance of the Hochschild complex under affine transformations, we instead exploit the observation that the Hochschild complex carries an anti-involution, and that such anti-involutive deformations of the complex of polyvectors are essentially unique. We also establish analogous statements for deformation quantisations in ${\mathcal{C}}^{\mathrm{\infty }}$ and analytic settings.