I am interested in deformations of affine Poisson algebras, and so this is the setting in which I shall write out the elementary definitions involved. All algebras and vector spaces shall be over $\mathbb{C}$ although most of my questions make sense over any field. Any answers to any part of my question will be very welcome indeed, thanks in advance!

An affine Poisson algebra $A$ is a finitely generated, reduced, commutative algebra with a bracket $\{ \cdot, \cdot\} : A \otimes A \rightarrow A$ which is skew-symmetric, satisfies the Jacobi identity and such that for all $f \in A$ the map $\{f, \cdot\} : A \rightarrow A$ is a derivation.

I am aware of two broad notions of deformation. Filtered deformations arise when $A$ is also graded $A = \bigoplus_{i \geq 0} A^i$ and the Poisson bracket is homogeneous of some fixed degree $d$, meaning $\{ \cdot, \cdot\} : A^i \otimes A^j \rightarrow A^{i + j - d}$ for all $i, j$. If $B$ is a non-commutative, filtered algebra with $B = \bigcup_{i \geq 0} B_i$ such that $[B_i, B_j] \subseteq B_{i + j-d}$ then $\operatorname{gr} B$ inherits a Poisson structure by setting $$\{ f, g\} := [\overline{f}, \overline{g}] + B_{\deg f + \deg g - d - 1}$$ for homogeneous elements $f, g \in \operatorname{gr} B$ where $\overline{f}, \overline{g}$ are lifts of $f,g$ to $B$. If $\operatorname{gr} B \cong A$ as Poisson algebras then we say that *$B$ is a filtered deformation of $A$*.

A closely related notion of deformation involves star products. Suppose that $A$ is a Poisson algebra as above. Then a $\ast$-product is an associative $\mathbb{C}[[\hbar]]$-linear product on $A[[\hbar]]$ which is written
$$f \ast g = fg + \hbar B_1(f, g) + \hbar^2 B_2(f,g) + \cdots$$
where $f, g \in A$ and $\ast$ satisfies $f \ast g - g \ast f = \hbar\{f, g\} + \mathcal{O}(\hbar^2)$. The algebra $(A[[\hbar]], \ast)$ is then referred to as a *deformation quantisation of the Poisson algebra $(A, \{ \cdot, \cdot\})$*.

Kontsevich famously defined a $\ast$-product on the $\mathcal{C}^\infty(M)$ where $M$ is a Poisson manifold, thus showing that deformation quantisations always exist in this case. Smooth affine Poisson varieties are special cases of these and I believe that the ring of regular functions can be endowed with a $\ast$-product à la Kontsevich.

I hope everything that I've written thus far is clear and correct.

**Question 1**: what is the precise connection between these two definitions of quantisation?

**Subquestion 1a**: When $B_i(f,g) = 0$ for $i \gg 0$ we can define $f \ast g|_{\hbar = 1}$ and if this holds for all $(f,g) \in A\times A$ we can define a product $\ast_{\hbar = 1} : A\otimes A \rightarrow A$. Does Kontsevich's $\ast$-product have this vanishing property (when $\operatorname{Spec} A$ is smooth)?

**Subquestion 1b**: If $(A, \{\cdot,\cdot\})$ is graded then is there automatically a filtration on $(A, \ast_{\hbar = 1})$ such that the associated graded is $(A, \{\cdot, \cdot\})$?

**Subquestion 1c**: If the Poisson structure on $A$ is not graded then how is the $\hbar = 1$ product on $A$ related to the Poisson structure? Is there something more general that a filtered deformation coming from this construction and can it be axiomatised?

Kontsevich's $\ast$-product is a consequence of his formality theorem, and so the choice of $\ast$-product is only well-defined up to so-called gauge equivalence, but gauge equivalent products subtend isomorphic algebras.

**Question 2**: Is isomorphisms a strictly finer invariant than gauge equivalence?

This can be asked in two settings:

**Subquestion 2a**: If $(A[[\hbar]], \ast) \cong (A[[\hbar]], \ast')$ then does it follow that the structures are gauge equivalent?

**Subquestion 2b**: If $(A, \ast_{\hbar = 1}) \cong (A, \ast'_{\hbar = 1})$ then does it follow that the structures are gauge equivalent?

Finally I would like to know something about quantising Poisson subvarieties. First comes the general question:

**Question 3**: Is Kontsevich's quantisation functorial in any sense?

Now comes the more specific version of this question. Suppose that $I \unlhd A$ is a Poisson ideal, so that $A/ I$ is a Poisson algebra and the projection $A \rightarrow A/I$ is a Poisson map. Kontsevich's quantisation gives products $(A[[\hbar]], \ast)$ and $(A/I[[\hbar]], \ast')$.

**Subquestion 3a**: Do we have a surjection $(A[[\hbar]], \ast) \twoheadrightarrow (A/I[\hbar]], \ast')$?

And finally the $\hbar = 1$ version:

**Subquestion 3b**: Do we have a surjection $(A, \ast_{\hbar = 1}) \twoheadrightarrow (A/I, \ast'_{\hbar=1})$?

Again, thanks in advance for any comments, answers, advice... anything at all really.