Scalar structure-sheaf cohomology of ${\mathbb{P}}^2$

The degree-zero exponents satisfy $m_0+m_1+m_2=0$. If $N(m)=\mathrm{\varnothing }$, all three exponents are nonnegative, so their sum is zero only for $m=(0,0,0)$. If $N(m)=\{0,1,2\}$, all three are negative and their sum is negative, impossible. Thus every other mode has nonempty proper negative set, one of the six sets $\{0\},\{1\},\{2\},\{0,1\},\{0,2\},\{1,2\}$. The least vertex outside these sets is respectively $1,0,0,2,1,0$.

For each nonzero mode choose its least $j\mathrm{\notin }N(m)$. On an increasing output index $I$ of size $p$, put $(h_{m}c{)}_I=0$ if $j\in I$ and otherwise put $(h_{m}c{)}_I=(-1{)}^{\mathrm{\#}\{i\in I:i<j\}}{c}_{I\cup \{j\}}$. For an input index $I$ of size $p+1$ with $j\mathrm{\notin }I$, the term of $h_m\delta c$ deleting $j$ has the same sign as the insertion sign, hence contributes $+c_I$; for every $i\in I$, the term deleting $i$ in $h_m\delta c$ and the term deleting $i$ in $\delta h_{m}c$ have opposite signs because the order of inserting $j$ and deleting $i$ is reversed. If $j\in I$, then $h_m\delta c$ vanishes and the only nonzero term in $\delta h_{m}c$ deletes $j$; its two signs agree and give $+c_I$. Terms deleting a vertex of $N(m)$ vanish because the resulting index does not contain $N(m)$. Therefore $\delta h_m+h_m\delta =\mathrm{i}\mathrm{d}$ in every degree for all six possible $N(m)$.

By [F2], the zero mode of the structure sheaf is a scalar constant on each nonempty intersection. Its Čech complex is $C^0={\mathbb{C}}^3$, $C^1={\mathbb{C}}^3$, $C^2=\mathbb{C}$, with maps ${\delta }^0(c_0,c_1,c_2)=(c_1-c_0,c_2-c_0,c_2-c_1)$ and ${\delta }^1(a_{01},a_{02},a_{12})=a_{12}-a_{02}+a_{01}$. The kernel of ${\delta }^0$ consists exactly of $(c,c,c)$, hence is $\mathbb{C}$. The image of ${\delta }^0$ is the plane $a_{02}=a_{01}+a_{12}$, which is exactly the kernel of ${\delta }^1$; ${\delta }^1$ is onto because $(0,0,1)$ maps to $1$. Thus this mode has $H^0=\mathbb{C}$ and $H^1=H^2=0$.

The modewise formula defines a homotopy on actual holomorphic cochains: by [F3], each output component is a sum of subseries grouped by at most the three choices of $j\mathrm{\notin }I$, each retained mode has no negative exponent at the added coordinate, and the subseries converge locally uniformly on $U_I$. Consequently their sums are holomorphic on $U_I$, including across the added coordinate hyperplane, and the coefficient identity in step 2.1 is an identity of analytic cochains.

Every nonzero mode is contractible by steps 1.1, 2.1 and 3.1, while the zero mode has the cohomology computed in step 2.2. The unique normally convergent coefficient expansions and their cochain homotopies make this coefficientwise computation valid for holomorphic sections, and [F3] identifies the resulting fixed-cover Čech cohomology with analytic sheaf cohomology. Hence the displayed sheaf-cohomology values follow.

Apply [F4] to the trivial line bundle $E={\mathcal{O}}_X$. Its global smooth Dolbeault complex has the same cohomology as the sheaf groups computed in step 4.1, giving the asserted Dolbeault-cohomology values.