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Compact coherent cohomology can be finite and nonzero

Example

Assume full AC. Let $P^{1} (C)$ have standard charts $U_{0} = {Z_{0} \neq 0}$ and $U_{1} = {Z_{1} \neq 0}$ , with coordinates $z = Z_{1} / Z_{0}$ on $U_{0} ≅ C$ and $w = Z_{0} / Z_{1} = 1 / z$ on $U_{1} ≅ C$ . Let $O (- 2)$ be the projective twisting line bundle with local frames satisfying $e_{1} = z^{- 2} e_{0}$ on $U_{0} \cap U_{1} ≅ C^{*}$ . Then

H^{1} (P^{1} (C), O (- 2)) ≅ C .

Thus finite-dimensionality of compact coherent cohomology does not imply vanishing of positive-degree cohomology.

Example references

Jean-Pierre Demailly, Complex Analytic and Differential Geometry, Chapter IX §§1.B and 4.D

Jeremy Orloff, MIT 18.04 Topic 7: Taylor and Laurent Series

Facts & Assumptions

Given: Full AC; the projective line with its two standard charts; and the line bundle $O (- 2)$ with the stated transition rule.

[F1]

The compact projective coherent-finiteness theorem asserts finite-dimensional Hausdorff cohomology for coherent analytic modules on $P^{1}$ (Finite-dimensional coherent cohomology on compact projective space); the calculation below determines the exact nonzero group.

[F2]

The local frames of $O (k)$ satisfy $e_{j}^{(k)} = (Z_{j} / Z_{i})^{k} e_{i}^{(k)}$ on the standard chart overlaps (Projective twisting line bundles by coordinate transitions).

[F3]

A domain is Hartogs pseudoconvex when $- \log δ_{Ω}$ is plurisubharmonic, where $δ_{Ω}$ is the equal-radius polydisc boundary function; the whole-space convention makes $C$ Hartogs pseudoconvex. On such domains the scalar holomorphic structure sheaf has zero positive-degree sheaf cohomology (Plurisubharmonic exhaustions and Hartogs pseudoconvexity, The equal-radius polydisc boundary function, Scalar holomorphic sheaf acyclicity on pseudoconvex domains).

[F4]

A $C^{2}$ function on a domain is plurisubharmonic when its Levi form is nonnegative (The Levi form and strict plurisubharmonicity, The $C^{2}$ Levi criterion for plurisubharmonicity).

[F5]

The ordered two-set cover computes derived cohomology by its Čech complex when the sheaf is acyclic on the two sets and their overlap (Leray acyclic-cover comparison, Canonical map from fixed-cover Čech to sheaf cohomology).

[F6]

Every holomorphic function on $C^{*}$ has a unique Laurent decomposition $f (z) = g (z) + a z^{- 1} + z^{- 2} h (1 / z)$ with $g, h$ entire and $a \in C$ (Laurent decomposition of holomorphic functions on the punctured plane).

[A1]

Full AC is the set-choice postulate and is included in the local Oka and scalar-acyclicity inputs (The Axiom of Choice, Oka coherence and local coherent free resolutions, Scalar holomorphic sheaf acyclicity on pseudoconvex domains).

[F7]

Under the Oka interface, the structure sheaf is coherent; a line bundle is locally free of rank one and therefore is a coherent analytic module (Oka coherence and local coherent free resolutions, Projective twisting line bundles by coordinate transitions).

Proof

1.1A1F2F3F4F7givenconstruct

By [F7], $O (- 2)$ is coherent, and its displayed frames trivialize it on $U_{0}$ and $U_{1}$ . The domain $U_{0} ≅ C$ is Hartogs pseudoconvex by the whole-space convention. On $U_{0} \cap U_{1} ≅ C^{*}$ , the equal-radius disk boundary function is $δ (z) = ∣ z ∣$ , so $- \log δ (z) = - \log ∣ z ∣$ is harmonic and has zero Levi form; it is plurisubharmonic by [F4], hence the overlap is Hartogs pseudoconvex. The same argument applies to $U_{1} ≅ C$ . By [F3], $O$ and thus $O (- 2)$ are acyclic on every nonempty member and overlap.

1.2F2F5givenconstruct

By [F5], $H^{1} (P^{1}, O (- 2))$ is the degree-one cohomology of the ordered Čech complex for $(U_{0}, U_{1})$ . A degree-one cochain is $f (z) e_{0}$ with $f \in O (C^{*})$ . A degree-zero cochain is $(a (z) e_{0}, b (w) e_{1})$ with $a, b \in O (C)$ . Using $e_{1} = z^{- 2} e_{0}$ , its coboundary on the overlap is $(z^{- 2} b (1 / z) - a (z)) e_{0}$ . Therefore the degree-one cohomology is the quotient of $O (C^{*})$ by the subspace $O (C) + {z^{- 2} b (1 / z) : b \in O (C)}$ .

2.1F6step 1.2algebra

For every $f \in O (C^{*})$ , [F6] writes it uniquely as $f (z) = g (z) + a z^{- 1} + z^{- 2} h (1 / z)$ with $g, h$ entire. The first and last terms lie in the coboundary subspace of step 1.2, so every quotient class is represented by $a z^{- 1}$ . If $a z^{- 1}$ itself lies in that subspace, its Laurent coefficient of degree $- 1$ is zero because an entire $a (z)$ has only nonnegative powers and $z^{- 2} b (1 / z)$ has only powers at most $- 2$ ; uniqueness in [F6] forces $a = 0$ . Thus the quotient is one-dimensional with basis class $[z^{- 1} e_{0}]$ .

3.1F1step 1.2step 2.1∎

The quotient computation in step 2.1 and the Leray identification in step 1.2 give $H^{1} (P^{1} (C), O (- 2)) ≅ C$ . In particular, [F1] supplies the compact coherent finiteness conclusion while this calculation exhibits a nonzero positive-degree group. This is a direct analytic Čech calculation using one-variable Laurent series; it invokes neither GAGA nor a positive-twist theorem.

Compact coherent cohomology can be finite and nonzero

Example

Example references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources