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Projective Dolbeault comparison and scalar structure-sheaf cohomology

Statement

Assume the full Axiom of Choice. Let $r \in {0, 1, 2, 3}$ , let $X = P^{r} (C) = (C^{r + 1} ∖ {0}) / C^{\times}$ with its standard quotient complex-manifold structure, and let $E \to X$ be a holomorphic vector bundle of finite rank. Write $O (E)$ for its sheaf of holomorphic sections, and let $H^{q} (X, F)$ denote the $q$ th right-derived global-sections group of an abelian sheaf $F$ on $X$ . For $0 \leq q \leq r$ , let $A^{0, q} (E)$ be the sheaf of smooth sections of $Λ^{0, q} T^{*} X \otimes_{C} E$ , where $Λ^{0, q} T^{*} X$ is the $q$ th exterior power of the antiholomorphic cotangent subbundle. In a holomorphic local frame of $E$ , define ${\overset{ˉ}{\partial}}_{E}$ by applying the scalar $\overset{ˉ}{\partial}$ to each coefficient form; holomorphic transition matrices make this definition independent of frame, and ${\overset{ˉ}{\partial}}_{E}^{2} = 0$ . Set $A^{0, q} (E) = 0$ for $q > r$ and define $H_{\overset{ˉ}{\partial}}^{q} (X, E) : = \frac{\ker ({\overset{ˉ}{\partial}}_{E} : Γ (X, A^{0, q} (E)) \to Γ (X, A^{0, q + 1} (E)))}{im ({\overset{ˉ}{\partial}}_{E} : Γ (X, A^{0, q - 1} (E)) \to Γ (X, A^{0, q} (E)))},$ where $Γ (X, F)$ denotes the group of sections of $F$ over $X$ and the denominator is zero for $q = 0$ . For every $q \geq 0$ , the canonical comparison induced by the smooth Dolbeault resolution is an isomorphism $H^{q} (X, O (E)) ≅ H_{\overset{ˉ}{\partial}}^{q} (X, E) .$ For the trivial line bundle $E = O_{X}$ and $0 \leq r \leq 3$ , this gives $H^{0} (P^{r}, O) = C, H^{q} (P^{r}, O) = 0 (q > 0) .$ The scalar cohomology conclusion is obtained by the direct Laurent–Čech calculation on the standard projective cover; no GAGA theorem, Kähler Hodge theorem in dimensions above one, or arbitrary-coherent projective vanishing is assumed.

Statement references

Lebl, Tasty Bits of Several Complex Variables, §4.4

MIT OpenCourseWare 18.117 lecture notes

Dervan–Kuang, Complex Manifolds, §§3–4

Serre, Géométrie algébrique et géométrie analytique, §13

Facts & Assumptions

Given: Full AC; $r \in {0, 1, 2, 3}$ ; the complex manifold $X = P^{r} (C)$ ; a finite-rank holomorphic vector bundle $E \to X$ ; and an integer $q \geq 0$ .

[A1]

Full AC means every family of nonempty sets has a choice function (The Axiom of Choice).

[F1]

For $r \geq 1$ , $X$ is the quotient of $C^{r + 1} ∖ {0}$ by nonzero scalar multiplication and has standard holomorphic charts $U_{i} = {Z_{i} \neq 0}$ with coordinates $Z_{j} / Z_{i}$ ; $P^{0}$ is a point (Complex projective space and its holomorphic charts).

[F2]

A holomorphic vector bundle is locally a product with holomorphic transition matrices, and its holomorphic-section sheaf consists of the local holomorphic sections (Complex manifolds, holomorphic vector bundles and Hermitian metrics).

[F3]

Complex differential forms split into $(p, q)$ -types; $\overset{ˉ}{\partial}$ raises $q$ by one and satisfies ${\overset{ˉ}{\partial}}^{2} = 0$ (Bigraded complex forms and the Dolbeault operators).

[F4]

On nested polydiscs, a smooth $\overset{ˉ}{\partial}$ -closed $(p, q)$ -form with $q > 0$ on the larger one has a smooth primitive on the smaller one under full AC (The local Dolbeault lemma on nested polydiscs).

[F5]

A $C^{1}$ function on a domain in $C^{n}$ satisfying the coordinate Cauchy–Riemann equations is holomorphic (For $C^{1}$ functions, holomorphy, complex linearity of the real derivative, and the Cauchy–Riemann system agree).

[F6]

Under AC, any sheaf of modules over the smooth-function sheaf on a smooth manifold is acyclic for global sections on every open subset (Sheaves of smooth-function modules are cohomologically acyclic).

[F7]

An exact resolution by $Γ$ -acyclic objects computes the right-derived functor when the initial object and every resolution cycle lie in the domain of the supplied injective-resolution data; DC is the choice hypothesis (The acyclic-resolution theorem for right derived functors).

[F8]

Under AC, $H^{q} (X, F)$ is the right-derived global-sections group for each abelian sheaf $F$ on $X$ , using supplied injective-resolution data defined on the category of all such sheaves (Sheaf cohomology as right derived global sections).

[F9]

In ZF, full AC implies DC (AC implies DC implies countable choice).

[F10]

The local twisting-bundle definition identifies $O_{P^{r}} (0)$ with the trivial line bundle and its holomorphic-section sheaf with the structure sheaf (Projective twisting line bundles by coordinate transitions).

[F11]

For $0 \leq r \leq 3$ , $H^{0} (P^{r}, O) = C$ and $H^{q} (P^{r}, O) = 0$ for every $q > 0$ (Laurent–Čech cohomology of the projective coordinate cover).

Proof

technique · direct

1.1A1F1F2F3F5F8givenconstructalgebra

If $r = 0$ , then $X$ is a point, so a sheaf on $X$ is just its group of sections, global sections is exact, and every positive sheaf-cohomology group vanishes. The degree-zero Dolbeault complex is the identity on the fiber $E$ , so the comparison is the identity by the derived-global-sections definition [F8]. Assume now $r \geq 1$ . On a coordinate polydisc with holomorphic frame $s_{1}, \dots, s_{n}$ of $E$ , write an $E$ -valued form as $\sum_{a = 1}^{n} α_{a} \otimes s_{a}$ and define ${\overset{ˉ}{\partial}}_{E} (\sum_{a} α_{a} \otimes s_{a}) = \sum_{a} \overset{ˉ}{\partial} α_{a} \otimes s_{a}$ . If the frame changes by a holomorphic matrix $G$ , then $\overset{ˉ}{\partial} (G α) = (\overset{ˉ}{\partial} G) \land α + G \overset{ˉ}{\partial} α = G \overset{ˉ}{\partial} α$ because every entry of $G$ is holomorphic and hence satisfies the Cauchy–Riemann equations [F5]. Thus the local operators agree on overlaps; by [F3], their square is zero.

2.1A1F2F4F5step 1.1construct

In a holomorphic frame, under full AC [A1], the local Dolbeault lemma [F4] applies componentwise to every closed form of positive antiholomorphic degree and gives exactness at that degree on a neighborhood of each point. At degree zero, a smooth coefficient killed by $\overset{ˉ}{\partial}$ satisfies the coordinate Cauchy–Riemann equations, so [F5] makes it holomorphic. Consequently the sheaf resolution $0 ⟶ O (E) ⟶ A^{0, 0} (E) \overset{{\overset{ˉ}{\partial}}_{E}}{\to} \dots \overset{{\overset{ˉ}{\partial}}_{E}}{\to} A^{0, r} (E) ⟶ 0$ is exact: its degree-zero kernel is exactly the holomorphic-section sheaf, and every positive-degree kernel is locally the preceding image.

3.1A1F6step 2.1

Each $A^{0, j} (E)$ is a sheaf of modules over smooth functions, by pointwise scalar multiplication; hence it is $Γ (X, -)$ -acyclic by [F6] under [A1].

4.1A1F7F8F9step 2.1step 3.1given

Every cycle in the exact resolution of step 2.1 is an abelian sheaf on $X$ , so it lies in the all-abelian-sheaves domain of the injective-resolution datum in [F8]. By [F9], full AC [A1] supplies the DC required by the acyclic-resolution theorem [F7]. Applying [F7] to that resolution and the global-sections functor gives, canonically, $H^{q} (X, O (E)) ≅ H^{q} (Γ (X, A^{0, ∙} (E))) = H_{\overset{ˉ}{\partial}}^{q} (X, E)$ for every $q \geq 0$ .

5.1F10F11givenalgebra∎

For $E = O_{X}$ , [F10] identifies its holomorphic-section sheaf with $O$ . Then [F11] supplies $H^{0} (X, O) = C$ and $H^{q} (X, O) = 0$ for $q > 0$ in every stated dimension, as claimed.

Projective Dolbeault comparison and scalar structure-sheaf cohomology

Statement

Statement references

Facts & Assumptions

Proof

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