Laurent–Čech cohomology of the projective coordinate cover

Assume $r\ge 1$ and fix a nonempty index set $I\subseteq \{0,\dots ,r\}$. Choose $i_0\in I$ and use $w_j=Z_j\mathrm{/}{Z}_{i_0}$ for $j\mathrm{\ne }{i}_0$ on $U_I$. The coordinates with $j\in I\setminus \{i_0\}$ are nonzero and all remaining coordinates range over $\mathbb{C}$, so $U_I\cong {\mathbb{C}}^{r+1-\mathrm{\mid }I\mathrm{\mid }}\times ({\mathbb{C}}^{\ast }{)}^{\mathrm{\mid }I\mathrm{\mid }-1}$; this product is open and connected. Write $b=\mathrm{\mid }I\mathrm{\mid }-1$.

The local Dolbeault lemma [F8], applied coefficientwise in a holomorphic frame on coordinate polydiscs in $U_I$, gives local exactness in every positive degree. In degree zero, a smooth function killed by $\bar{\mathrm{\partial }}$ satisfies the Cauchy–Riemann equations and is holomorphic by [F9]. By [F7] the operators form a complex, so $0\to \mathcal{O}\to {\mathcal{A}}^{0,0}\stackrel{\bar{\mathrm{\partial }}}{\to }\cdots \stackrel{\bar{\mathrm{\partial }}}{\to }{\mathcal{A}}^{0,r}\to 0$ is an exact sheaf resolution. The whole-polydisc statement [F10] also gives positive-degree exactness on each coordinate polydisc. Each ${\mathcal{A}}^{0,j}$ is a module over smooth functions, hence is $\mathrm{\Gamma }(U_I,-)$-acyclic by [F11]. The resolution cycles lie in the supplied injective-data domain [F13], and [A1] supplies the DC required by [F12]. Applying [F12] identifies $H^q(U_I,\mathcal{O})$ with the cohomology of the global Dolbeault complex on $U_I$.

For a section of $\mathcal{O}(k)$ on $U_I$, write its coefficient in frame $e_{i_0}^{(k)}$ as a holomorphic function $f$ on ${\mathbb{C}}^a\times ({\mathbb{C}}^{\ast }{)}^b$, where $a=r+1-\mathrm{\mid }I\mathrm{\mid }$ and $b=\mathrm{\mid }I\mathrm{\mid }-1$. By [F17], $f$ is continuous and separately holomorphic. Iterating the one-variable Cauchy expansions on the $\mathbb{C}$ factors and Laurent expansions on the ${\mathbb{C}}^{\ast }$ factors gives a mixed series indexed by $m_j\in \mathbb{N}$ for coordinates that may vanish and $m_j\in \mathbb{Z}$ for the nonzero coordinates. To prove holomorphy of each coefficient in the remaining variables, fix the Laurent contour radii and a closed parameter polydisc compactly contained in the remaining coordinates. For each fixed point $\zeta$ of the contour product, apply the iterated Cauchy formula [F15] to the remaining variables. Since $f$ is bounded on the compact product of the contours and that polydisc, the Taylor coefficient estimates are uniform in $\zeta$. On a smaller concentric polydisc, the Taylor series therefore converges absolutely and uniformly in $\zeta$. The estimate $\mathrm{\mid }{\int }_{\mathrm{\Gamma }}g\mathrm{\mid }\le L_{\mathrm{\Gamma }}{\sup }_{\mathrm{\Gamma }}\mathrm{\mid }g\mathrm{\mid }$ allows termwise integration in the already fixed contour order, so the Laurent coefficient has a convergent power series in all remaining variables. Thus it is holomorphic. Define $m_{i_0}=k-{\sum }_{j\mathrm{\ne }{i}_0}{m}_j$ and write $Z^m$ for the homogeneous Laurent monomial section whose coefficient in frame $e_{i_0}^{(k)}$ is ${\prod }_{j\mathrm{\ne }{i}_0}(Z_j\mathrm{/}{Z}_{i_0}{)}^{m_j}$; its total degree is ${\sum }_i{m}_i=k$. Every negative exponent is among coordinates indexed by $I$.

If $b=0$, then $U_I\cong {\mathbb{C}}^r$ and ${\delta }_{U_I}=+\mathrm{\infty }$ by [F5]. If $b>0$, an equal-radius polydisc of radius $s$ around $w$ is contained in $U_I$ exactly when $s\le {\min }_{j\in I\setminus \{i_0\}}\mathrm{\mid }{w}_j\mathrm{\mid }$: smaller radii cannot reach a forbidden coordinate hyperplane, while any larger radius includes zero in a coordinate disk attaining the minimum. Hence ${\delta }_{U_I}(w)={\min }_j\mathrm{\mid }{w}_j\mathrm{\mid }$ and $-\log {\delta }_{U_I}={\max }_j(-\log \mathrm{\mid }{w}_j\mathrm{\mid })$. Each $-\log \mathrm{\mid }{w}_j\mathrm{\mid }$ is harmonic on ${\mathbb{C}}^{\ast }$ because it is locally the real part of a holomorphic logarithm. Their finite maximum is continuous and subharmonic on each complex line: at a point choose an index attaining the maximum, apply the submean inequality for that harmonic function, and bound its boundary values by the maximum. Thus $-\log {\delta }_{U_I}$ is plurisubharmonic by [F4], so $U_I$ is Hartogs pseudoconvex by [F5].

The domain $U_I$ has complex dimension $r$, so [F6] makes its global Dolbeault cohomology zero for $1\le q\le r$; for $q>r$ the complex has no terms in degree $q$, hence its cohomology is zero as well. The frame $e_{i_0}^{(k)}$ trivializes $\mathcal{O}(k)$ on $U_I$, so its restricted holomorphic-section sheaf is isomorphic to $\mathcal{O}$ and has the same positive-degree sheaf cohomology.

Let $K$ be a compact product with $\mathrm{\mid }{w}_j\mathrm{\mid }\le S_j$ on each $\mathbb{C}$ coordinate and ${\rho }_j\le \mathrm{\mid }{w}_j\mathrm{\mid }\le R_j$ on each ${\mathbb{C}}^{\ast }$ coordinate. Choose $S_j^{\mathrm{\prime }}>S_j$, $R_j^{\mathrm{\prime }}>R_j$ and $0<{\rho }_j^{\mathrm{\prime }}<{\rho }_j$; the corresponding Cauchy and Laurent circles stay in the domain. The iterated coefficient-integral bounds give geometric factors $(S_j\mathrm{/}{S}_j^{\mathrm{\prime }}{)}^{m_j}$ for a Taylor coordinate, $(R_j\mathrm{/}{R}_j^{\mathrm{\prime }}{)}^{m_j}$ for a nonnegative Laurent exponent, and $({\rho }_j^{\mathrm{\prime }}\mathrm{/}{\rho }_j{)}^{\mathrm{\mid }{m}_j\mathrm{\mid }}$ for a negative Laurent exponent. The remaining factor is bounded by the supremum of $f$ on the finite product of chosen circles, times fixed contour-length factors. There are finitely many choices of signs, and each product of geometric series converges; hence the mixed series converges absolutely and uniformly on $K$.

Define $C^p={\prod }_{\mathrm{\mid }I\mathrm{\mid }=p+1}\mathcal{O}(k)(U_I)$, with the ordered Čech differential $(\delta c{)}_{i_0\cdots i_{p+1}}={\sum }_{\nu =0}^{p+1}(-1{)}^{\nu }{c}_{i_0\cdots \widehat{i_{\nu }}\cdots i_{p+1}}{\mathrm{\mid }}_{U_{i_0\cdots i_{p+1}}}$. For a multi-index $m$ of total degree $k$, set $N(m)=\{i:m_i<0\}$. By step 1.3, its coefficient occurs in the component indexed by $I$ exactly when $N(m)\subseteq I$. The frame transition rule in [F2] changes the local coefficient by $(Z_i\mathrm{/}{Z}_j{)}^k$, exactly the change between the two local expressions of the same degree-$k$ monomial $Z^m$; restrictions consequently preserve $m$. Thus the differential acts on each coefficient by the finite alternating-subset complex whose degree-$p$ entries are scalars indexed by subsets $I$ of size $p+1$ containing $N(m)$.

Give the cover $U_0,\dots ,U_r$ its natural order. Step 2.2 proves that every nonempty finite intersection is acyclic for each coefficient sheaf $\mathcal{O}(k)$ under consideration. Therefore [F14] identifies its analytic sheaf cohomology with the fixed-cover Čech cohomology of the ordered cover.

Each mixed coefficient is given by its fixed iterated Cauchy or Laurent contour integral. Applying these same integrals in the same order to the uniformly convergent series on a compact polyannulus recovers that coefficient; the uniqueness of the one-variable Cauchy and Laurent coefficients therefore proves uniqueness of the mixed coefficients.

If $N(m)$ is nonempty and proper, choose the least $j\mathrm{\notin }N(m)$. For an increasing output index $I$ of size $p$ containing $N(m)$, define $(h_{m}c{)}_I=0$ when $j\in I$, and otherwise define $(h_{m}c{)}_I=(-1{)}^{\mathrm{\#}\{i\in I:i<j\}}{c}_{I\cup \{j\}}$; this maps degree $p$ to degree $p-1$ for $p\ge 1$. Put $C^{-1}=0$ and define $h_m:C^0\to C^{-1}$ to be zero. In degree zero, if $\mathrm{\mid }N(m)\mathrm{\mid }>1$, then $C^0$ for this mode is zero. If $N(m)=\{i\}$, the only possibly nonzero degree-zero coefficient is $c_{\{i\}}$. For the chosen $j\mathrm{\ne }i$, the face of $\{i,j\}$ deleting $i$ is indexed by $\{j\}$, which omits $N(m)$ and hence has coefficient zero. The face deleting $j$ has sign $ϵ=(-1{)}^{\mathrm{\#}\{k\in \{i\}:k<j\}}$, so $(\delta c{)}_{\{i,j\}}=ϵc_{\{i\}}$ in increasing order and $(h_m\delta c{)}_{\{i\}}={ϵ}^2{c}_{\{i\}}=c_{\{i\}}$. Since $\delta h_{m}c=0$, the contraction identity holds in degree zero, including its minimal index. To check the identity in positive degrees at an input index $I$ of size $p+1$, first suppose $j\mathrm{\notin }I$. In $h_m\delta c$, the face obtained by deleting $j$ from $I\cup \{j\}$ has sign $(-1{)}^{\mathrm{\#}\{i\in I:i<j\}}$; multiplication by the same sign in $h_m$ leaves $c_I$ with coefficient $+1$. For each $i\in I$, the other term obtained by deleting $i$ occurs also in $\delta h_{m}c$, and its two signs differ by one because inserting $j$ and deleting $i$ reverse order; those terms cancel pairwise. If $j\in I$, then $h_m\delta c$ is zero at $I$, all terms of $\delta h_{m}c$ vanish except the face deleting $j$, and its two equal signs again multiply to $+1$, leaving $c_I$. Faces that delete an element of $N(m)$ have zero coefficient because the resulting index no longer contains $N(m)$. Thus $\delta h_m+h_m\delta =\mathrm{i}\mathrm{d}$ in every degree, including the minimal index $I=N(m)$.

For $k=0$, the only multi-index with $N(m)=\mathrm{\varnothing }$ is $m=(0,\dots ,0)$; its coefficient complex is the full simplex complex. Its degree-zero kernel is the constant tuples $\mathbb{C}$. For positive degree define $h{c}_I=0$ when $0\in I$ and $h{c}_I=c_{I\cup \{0\}}$ when $0\mathrm{\notin }I$; inserting the least vertex $0$ gives positive sign, and the insertion/deletion identity proved in step 3.3 gives $\delta h+h\delta =\mathrm{i}\mathrm{d}$. Every other degree-zero mode has nonempty proper $N(m)$ and is contracted by step 3.3. For $k=-1$ and $r\ge 1$, no mode has empty or full negative set: all-nonnegative exponents cannot sum to $-1$, and all-negative exponents sum to at most $-(r+1)$. Thus step 3.3 contracts every mode and degree-zero cohomology is zero. For $k=-r-1$, the unique mode with full negative set is $m=(-1,\dots ,-1)$: writing each $m_i=-1-n_i$ with $n_i\ge 0$, the total-degree equation gives ${\sum }_i{n}_i=0$, hence every $n_i=0$. This mode occurs only in top degree $r$, has no incoming or outgoing Čech term, and contributes one copy of $\mathbb{C}$. All other modes have nonempty proper negative set and contract.

The coefficient homotopy of step 3.3 defines an analytic cochain homotopy, not just a formal one. In an output component indexed by $I$, group terms by the finitely many possible selected vertices $j\mathrm{\notin }I$. Each retained mode has $N(m)\subseteq I$ and hence $m_j\ge 0$. For any compact product in $U_I$, choose outer Cauchy circles beyond its bounded coordinates and inner Laurent circles below the positive minimum of each nonzero coordinate. The estimates of step 2.3 then give absolute locally uniform convergence of each retained subseries on that compact set. Each partial sum is holomorphic there, so its limit is holomorphic by [F18]; the nonnegative $m_j$ powers extend the subseries across the coordinate hyperplane $Z_j=0$. Thus $h$ maps holomorphic Čech cochains to holomorphic Čech cochains, and the coefficient identity of step 3.3 holds for the analytic sums.

The Čech computation in step 4.1 is therefore valid for analytic sections. By the comparison of step 3.1 it gives $H^0(P^r,\mathcal{O})=\mathbb{C}$ and $H^q(P^r,\mathcal{O})=0$ for $q>0$, together with $H^0(P^r,\mathcal{O}(-1))=0$ and $H^r(P^r,\mathcal{O}(-r-1))\cong \mathbb{C}$ for $1\le r\le 3$. For $r=0$, $P^0$ is a point, so its structure-sheaf global sections are $\mathbb{C}$ and its higher sheaf cohomology groups vanish.