Noncommutative holomorphic functional calculus, affine and projective spaces from $\operatorname{NC}$-complete algebras

The paper is devoted to a noncommutative holomorphic functional calculus and its application to noncommutative algebraic geometry. A description is given for the noncommutative (infinite-dimensional) affine spaces $\mathbb{A}_q^{\mathbf {x}}$, $1\leq q\leq \infty$, $\mathbf {x}=(x_{\iota })_{\iota \in \Xi }$, and for the projective spaces $\mathbb{P}_q^n$ within Kapranov's model of noncommutative algebraic geometry based on the sheaf of formally-radical holomorphic functions of elements of a nilpotent Lie algebra and on the related functional calculus. The obtained result for $q=\infty$ generalizes Kapranov's formula in the finite dimensional case of $\mathbb{A}_q^n$. The noncommutative scheme $\mathbb{P}_q^n$ corresponds to the graded universal enveloping algebra $\mathcal {U}(\mathfrak{g}_q(\mathbf {x}))$ of the free nilpotent Lie algebra of index $q$ generated by $\mathbf {x}=(x_0,\dots ,x_n)$ with $\deg (x_i)=1$, $0\leq i\leq n$. A sheaf construction $B\big (\mathbb{P}^n,f_q,\mathcal {O}(-2),\dots ,\mathcal {O}(-q)\big )$ is suggested, in terms of the twisted sheaves $\mathcal {O}(-2)$, $\dots$, $\mathcal {O}(-q)$ on $\mathbb{P}^n$ and the formal power series $f_q$, to restore the coordinate ring of $\mathbb{P}_q^n$ that is reduced to $\mathcal {U}(\mathfrak{g}_q(\mathbf {x}))$. Finally, the related cohomology groups $H^i\big (\mathbb{P}_q^n$, $\mathcal {O}_q(d)\big )$, $i\geq 0$, are calculated.