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.