Failure of global generation for $\mathcal{O}(-1)$ and top cohomology of $\mathcal{O}(-r-1)$

For a degree-$-1$ mode, $N(m)$ cannot be empty because then every $m_i\ge 0$ and ${\sum }_i{m}_i\ge 0$; it cannot be all of $J:=\{0,\dots ,r\}$ because then ${\sum }_i{m}_i\le -(r+1)\le -2$. Thus every mode has nonempty proper negative support. If $\mathrm{\mid }N(m)\mathrm{\mid }\ge 2$, that mode occurs in no singleton-chart term of $C^0$. If $N(m)=\{i\}$, it occurs only in the $U_i$ term of $C^0$, while it occurs on $U_{ij}$ for any $j\mathrm{\ne }i$; the $0$-cocycle condition on that overlap forces its coefficient in the $U_i$ term to be zero because the $U_j$ term has no such mode. Hence every degree-zero Čech cocycle has zero coefficient in every mode. The normal-convergence and coefficient-uniqueness clauses of [F4] make this coefficientwise conclusion valid for analytic sections, and the fixed-cover comparison in [F4] gives $H^0(X,\mathcal{O}(-1))=0$.

The top Čech cochains have the single index set $J=\{0,\dots ,r\}$. Write $Z^m$ for the degree-$k$ monomial section whose coefficient in frame $e_i^{(k)}$ is ${\prod }_{\mathrm{\ell }\mathrm{\ne }i}(Z_{\mathrm{\ell }}\mathrm{/}{Z}_i{)}^{m_{\mathrm{\ell }}}$. A mode with $N(m)=J$ is absent from every codimension-one face, so it has zero coefficient in every top coboundary. If $\mathrm{\varnothing }\mathrm{\ne }N(m)\mathrm{\ne }J$, the convergent contraction in [F4] supplies a primitive. If $N(m)=\mathrm{\varnothing }$, all exponents are nonnegative, so $Z^m$ is holomorphic on every chart by [F3]. Set $b_{J\setminus \{0\}}=Z^m$ and all other components of $b\in C^{r-1}(k)$ to zero; the ordered differential gives $(\delta b{)}_J=(-1{)}^0{Z}^m=Z^m$. There are only finitely many such nonnegative exponent tuples of any fixed total degree $k$ (none for $k<0$). Their primitives therefore sum finitely; [F4] handles the normally convergent sums with nonempty proper negative support. Thus top cohomology has exactly the modes with $N(m)=J$.

For $k=-r-1$, if every $m_i<0$, write $m_i=-1-n_i$ with integers $n_i\ge 0$. The total-degree equation becomes $-(r+1)-{\sum }_i{n}_i=-(r+1)$, so ${\sum }_i{n}_i=0$ and every $n_i=0$. Thus the only surviving mode is $m=(-1,\dots ,-1)$, and its coefficient space is $\mathbb{C}$. The normal convergence and analytic cochain contraction in [F4] show that this coefficient quotient is the actual analytic Čech quotient. The comparison in [F4] now gives $H^r(X,\mathcal{O}(-r-1))\cong \mathbb{C}$.

Apply [F5] to both holomorphic line bundles $\mathcal{O}(-1)$ and $\mathcal{O}(-r-1)$. The canonical isomorphisms identify their global Dolbeault cohomology with the sheaf groups computed in steps 1.1 and 2.1, so the Dolbeault values are zero in degree zero for $\mathcal{O}(-1)$ and $\mathbb{C}$ in degree $r$ for $\mathcal{O}(-r-1)$.

By [F2], each fiber $\mathcal{O}(-1{)}_p$ is a one-dimensional complex vector space. Evaluation at any $p\in X$ is the zero map from $H^0(X,\mathcal{O}(-1))=0$ to this nonzero fiber, so it is not onto. This violates the defining fiberwise condition for global generation, proving that $\mathcal{O}(-1)$ is not globally generated.