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Coherent sheaves are acyclic on pseudoconvex intersections with local resolutions

Statement

Assume full AC and the local Oka interface stated below. Let $r \geq 1$ , let $M$ be a complex manifold of complex dimension $r$ , and let $V$ be an open subset of $M$ on which a coherent analytic module $F$ is defined. Let $D_{1}, \dots, D_{s}$ be relatively compact coordinate polydiscs in $M$ with nonempty intersection $V = ⋂_{i = 1}^{s} D_{i}$ . Suppose that for one index $i_{0}$ , a neighborhood of $\overline{D_{i_{0}}}$ in its chart lies in the domain of $F$ and carries an exact partial finite-rank free resolution of $F$ of length $2 r$ . Then $H^{q} (V, F ∣_{V}) = 0$ for every $q > 0$ . The result applies to a disconnected $V$ componentwise. The assumed local Oka interface is: under AC, on every finite-dimensional complex manifold $O_{M}$ is coherent, coherent modules are closed under kernels, cokernels and finite extensions, and for each coherent $F$ , point $x$ and finite integer $ℓ \geq 0$ , an exact partial finite-rank resolution $O^{p_{ℓ}} \to \dots \to O^{p_{0}} \to F \to 0$ exists on a chart neighborhood, exact at the module and all free terms except possibly the leftmost. No global free resolution is asserted.

Statement references

Jean-Pierre Demailly, Complex Analytic and Differential Geometry, Chapter IX §4.B

Yuan Liu, Cartan’s method and its applications in sheaf cohomology, §2.1

Facts & Assumptions

Given: Full AC; an $r$ -dimensional complex manifold; a coherent analytic module on a neighborhood of $V$ ; finitely many relatively compact coordinate polydiscs with intersection $V$ nonempty; and an exact length- $2 r$ partial free resolution on a neighborhood of one participating polydisc. The resolution is only assumed exact through the penultimate free term.

[A1]

Oka coherence and local partial resolutions: assume AC; on every finite-dimensional complex manifold $M$ , $O_{M}$ is coherent, coherent modules are closed under kernels, cokernels and finite extensions, and for every coherent $F$ , point $x$ and finite $ℓ \geq 0$ there is a chart neighborhood with an exact partial finite-rank free resolution $O^{p_{ℓ}} \to \dots \to O^{p_{0}} \to F \to 0$ , exact at the module and each free term except the leftmost. No global free resolution is asserted (Oka coherence and local coherent free resolutions).

[F1]

A continuous plurisubharmonic exhaustion on a domain can be regularized to a smooth strictly plurisubharmonic exhaustion under AC (Smooth strict plurisubharmonic regularization of a psh exhaustion).

[F2]

A bounded domain with a smooth plurisubharmonic exhaustion is Hartogs pseudoconvex (A smooth psh exhaustion gives Hartogs pseudoconvexity on bounded domains).

[F3]

A Hartogs-pseudoconvex domain has zero positive-degree derived cohomology for its holomorphic structure sheaf (Scalar holomorphic sheaf acyclicity on pseudoconvex domains, Positive-degree Dolbeault vanishing on pseudoconvex domains).

[F4]

If $m \geq 1$ and $W \subseteq R^{m}$ is nonempty and open and $G$ is any sheaf of abelian groups on $W$ , then $H^{j} (W, G) = 0$ for $j > m$ (Ordinary sheaf cohomology on Euclidean open sets vanishes above real dimension). This is ordinary cohomology, not a compact-support assertion.

[F5]

A short exact sequence of sheaves gives the long exact sequence of derived sheaf cohomology (Long exact sequence of sheaf cohomology).

[F6]

The coordinate polydisc $D = {∣ z_{j} - a_{j} ∣ < R_{j}}_{j = 1}^{r}$ has the smooth strictly plurisubharmonic exhaustion $ρ_{D} (z) = - \sum_{j = 1}^{r} \log (R_{j}^{2} - ∣ z_{j} - a_{j} ∣^{2})$ ; its Levi matrix is diagonal with entries $R_{j}^{2} / (R_{j}^{2} - ∣ z_{j} - a_{j} ∣^{2})^{2} > 0$ . Finite maxima preserve plurisubharmonicity, and holomorphic coordinate changes preserve it (Plurisubharmonic functions, The Levi form and strict plurisubharmonicity, The $C^{2}$ Levi criterion for plurisubharmonicity, Basic stability operations for plurisubharmonic functions, Holomorphic pullbacks of $C^{2}$ plurisubharmonic functions are plurisubharmonic).

Proof

1.1F6givenconstruct

Choose the chart of $D_{i_{0}}$ and write $V$ in those coordinates. For each $D_{i}$ , the function $ρ_{D_{i}}$ in [F6] is a smooth plurisubharmonic exhaustion of $D_{i}$ ; on $V$ it remains plurisubharmonic after composing with the holomorphic chart transition. The continuous function $ρ : = \max_{1 \leq i \leq s} ρ_{D_{i}}$ is plurisubharmonic by [F6]. If a sequence in $V$ leaves every compact subset of $V$ , it approaches the boundary of at least one of the finitely many $D_{i}$ , and the corresponding $ρ_{D_{i}}$ tends to $+ \infty$ . Thus $ρ$ is an exhaustion of $V$ .

2.1A1F1F2F3step 1.1

Each connected component $V_{α}$ is a bounded domain in the chart of $D_{i_{0}}$ and inherits the continuous plurisubharmonic exhaustion $ρ$ . Apply [F1] to obtain a smooth plurisubharmonic exhaustion on $V_{α}$ , then [F2] to conclude that $V_{α}$ is Hartogs pseudoconvex. By [F3], $H^{q} (V_{α}, O) = 0$ for $q > 0$ , hence $H^{q} (V_{α}, O^{p}) = 0$ for every finite rank $p$ .

3.1A1F5step 2.1given

Restrict the assumed partial resolution to $V_{α}$ and define $K_{- 1} = F ∣_{V_{α}}$ and $K_{j} = \ker (O_{V_{α}}^{p_{j}} \to O_{V_{α}}^{p_{j - 1}})$ for $1 \leq j \leq 2 r$ , with $K_{0} = \ker (O_{V_{α}}^{p_{0}} \to K_{- 1})$ . Exactness of the resolution through the penultimate free term gives short exact sequences $0 \to K_{j} \to O_{V_{α}}^{p_{j}} \to K_{j - 1} \to 0$ for $0 \leq j \leq 2 r$ , where at $j = 0$ the rightmost term is $K_{- 1}$ . By [A1] each $K_{j}$ is coherent, and by step 2.1 the middle free sheaf in each sequence is acyclic in positive degree.

4.1F4F5step 3.1

The long exact sequence [F5] for each short exact sequence in step 3.1 therefore gives, for every $q > 0$ , isomorphisms $H^{q} (V_{α}, K_{j - 1}) ≅ H^{q + 1} (V_{α}, K_{j})$ for $j = 0, 1, \dots, 2 r$ . Iterating yields $H^{q} (V_{α}, F) ≅ H^{q + 2 r + 1} (V_{α}, K_{2 r})$ . The chart identifies $V_{α}$ with an open subset of $C^{r} ≅ R^{2 r}$ , so [F4] makes the last group zero because $q + 2 r + 1 > 2 r$ .

5.1A1step 4.1algebra∎

The components of the open set $V$ are disjoint open subsets and are at most countable because the chart is second countable. Sheaves on their disjoint union are tuples of sheaves on the components; global sections are products, and products of abelian groups are exact under AC. Applying componentwise injective resolutions therefore gives $H^{q} (V, F) = \prod_{α} H^{q} (V_{α}, F) = 0$ for $q > 0$ by step 4.1.

Coherent sheaves are acyclic on pseudoconvex intersections with local resolutions

Statement

Statement references

Facts & Assumptions

Proof

Depends on

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Sources