Quantization of of an elliptic curve can be done in different ways.
In `C^*`

-algebraic version,
one can start with the `C^*`

-algebra of continuous functions on ordinary torus and by inserting a deformation parameter `\theta`

into the product obtain a deformed non-commutaive `C^*`

-algebra of functions on the quantum torus.

My question is:

Is there any natural way for deformation quantization of closed Riemann surfaces with higher genus in the above sense?