Complex structure on quantum-braided planes

We construct a quantum Dolbeault double complex $\oplus_{p,q}\Omega^{p,q}$ on the quantum plane $\Bbb C_q^2$. This solves the long-standing problem that the standard differential calculus on the quantum plane is not a $*$-calculus, by embedding it as the holomorphic part of a $*$-calculus. We show in general that any Nichols-Woronowicz algebra or braided plane $B_+(V)$, where $V$ is an object in an abelian $\Bbb C$-linear braided bar category of real type is a quantum complex space in this sense with a factorisable Dolbeault double complex. We combine the Chern construction on $\Omega^{1,0}$ in such a Dolbeault complex for an algebra $A$ with its conjugate to construct a canonical metric compatible connection on $\Omega^1$ associated to a class of quantum metrics, and apply this to the quantum plane. We also apply this to finite groups $G$ with Cayley graph generators split into two halves related by inversion, constructing such a Dolbeault complex $\Omega(G)$ in this case, recovering the quantum Levi-Civita connection for any edge-symmetric metric on the integer lattice with $\Omega(\Bbb Z)$ now viewed as a quantum complex structure. We also show how to build natural quantum metrics on $\Omega^{1,0}$ and $\Omega^{0,1}$ separately where the inner product in the case of the quantum plane, in order to descend to $\otimes_A$, is taken with values in an $A$-bimodule.