Analytic negative-twist vanishings for the positive real monad

Given: Full AC and the standard holomorphic hyperplane inclusions ${\mathbb{P}}^{r-1}\hookrightarrow {\mathbb{P}}^r$ for $1\le r\le 3$.

Under full AC, the batch-67 scalar cohomology theorem states that for every $0\le r\le 3$, $H^0({\mathbb{P}}^r,\mathcal{O})=\mathbb{C}$ and $H^q({\mathbb{P}}^r,\mathcal{O})=0$ for $q>0$. Projective Dolbeault comparison and scalar structure-sheaf cohomology

For a hyperplane inclusion $i:{\mathbb{P}}^{r-1}\hookrightarrow {\mathbb{P}}^r$, multiplication by its defining linear form gives the exact holomorphic sequences $0\to \mathcal{O}(-1)\to \mathcal{O}\to i_{\ast }{\mathcal{O}}_{{\mathbb{P}}^{r-1}}\to 0$ and $0\to \mathcal{O}(-2)\to \mathcal{O}(-1)\to i_{\ast }{\mathcal{O}}_{{\mathbb{P}}^{r-1}}(-1)\to 0$. Exact holomorphic hyperplane sequences for negative twists

For this closed inclusion, $i_{\ast }$ is exact because its stalk is the original stalk on $H={\mathbb{P}}^{r-1}$ and zero off $H$. The inverse-image functor $i^{-1}$ is exact and left adjoint to $i_{\ast }$, so $i_{\ast }$ sends injective sheaves to injective sheaves. Also $\mathrm{\Gamma }({\mathbb{P}}^r,i_{\ast }\mathcal{G})=\mathrm{\Gamma }(H,\mathcal{G})$. Applying $i_{\ast }$ to an injective resolution therefore gives the natural isomorphism $H^q({\mathbb{P}}^r,i_{\ast }\mathcal{G})\cong H^q(H,\mathcal{G})$ for every sheaf $\mathcal{G}$ on $H$.

A short exact sequence of sheaves gives the natural long exact sequence in the same sheaf cohomology groups. Long exact sequence of sheaf cohomology