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Finiteness on compact complex spaces

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

On a compact complex space, the cohomology of every coherent analytic sheaf is finite-dimensional in every degree. This theorem of H. Cartan and J.-P. Serre is the analytic counterpart of the finiteness of coherent cohomology on proper schemes, and it is the input that makes Serre's comparison of algebraic and analytic geometry work for arbitrary coherent analytic sheaves. The proof compares two finite coverings by small Stein pieces, one inside the other: both compute the cohomology, and the comparison map is compact, so Schwartz's theorem gives finiteness. The only new point is that finite intersections of the pieces, which may be embedded in different charts, are again acyclic; this follows from Theorem B applied on products of balls. The last section lists where the analytic results of this course are used in the proof of the comparison theorems.

We use Cartan's coherence theorem and complex spaces, Fréchet spaces of sections and Schwartz's theorem and Theorems A and B on Stein manifolds. Čech cohomology of a covering with acyclic finite intersections computes sheaf cohomology Stacks, Tag 01ET.

Basic references are [Cartan–Serre 1953], [Demailly] and [Serre 1956].

1. Stein pieces and their intersections

Let \(X\) be a complex space. A Stein piece of \(X\) is an open set \(U\subset X\) together with an isomorphism of \(U\) onto a closed complex subspace of a Stein manifold \(V\); for example \(U=A\cap V\), where \(A\subset\Omega\subset\mathbf C^N\) is a local model of \(X\) and \(V\subset\Omega\) is a ball, so that \(A\cap V\) is closed in \(V\).

Lemma 1.1. Let \(U_1,\ldots,U_k\) be Stein pieces of \(X\), embedded in Stein manifolds \(V_1,\ldots,V_k\) by \(e_i:U_i\to V_i\). Then \(e=(e_1,\ldots,e_k)\) embeds \(U=U_1\cap\cdots\cap U_k\) as a closed complex subspace of the Stein manifold \(V_1\times\cdots\times V_k\). Consequently, for every coherent sheaf \(\mathcal S\) on \(X\), \(H^j(U,\mathcal S)=0\) for all \(j\geq1\).

Proof. Closedness. Let \(e(x_\nu)\to(y_1,\ldots,y_k)\) in the product. Since \(e_1(U_1)\) is closed in \(V_1\), \(y_1=e_1(x)\) for some \(x\in U_1\), and \(x_\nu\to x\) because \(e_1\) is a homeomorphism onto its image; likewise \(x_\nu\to x'\in U_2\), and so on. As \(X\) is Hausdorff, all these limits coincide, so the limit point is \(e(x)\) with \(x\in U\).

Local structure. Let \(x\in U\). The components of \(e_2,\ldots,e_k\) are holomorphic functions on \(U\) near \(x\); through the embedding \(e_1\) they are, near \(e_1(x)\), restrictions of holomorphic functions \(h\) on an open set of \(V_1\), because a local model's structure sheaf is a quotient of the ambient one. Near \(e(x)\), the image of \(U\) is therefore the graph \(\{(y,h(y))\}\) of \(h\) over the local model \(e_1(U)\subset V_1\), which is the local model in \(V_1\times\mathbf C^{N_2+\cdots+N_k}\) defined by the ideal of \(e_1(U_1)\) together with the equations \(w-h(y)\); projection to \(V_1\) identifies it with \(e_1(U)\) as complex spaces. So \(e\) is an isomorphism of \(U\) onto a closed complex subspace of \(V_1\times\cdots\times V_k\), whose ideal sheaf, defined locally in this way, is coherent.

Vanishing. The product of Stein manifolds is Stein Plurisubharmonic functions and Stein manifolds, Proposition 2.2. By Cartan's coherence theorem and complex spaces, Theorem 3.2(3), \(e_*\mathcal S\) is coherent and \(H^j(U,\mathcal S)=H^j(V_1\times\cdots\times V_k,e_*\mathcal S)\), which vanishes for \(j\geq1\) by Theorems A and B on Stein manifolds, Theorem 4.1. \(\square\)

From acyclic intersections to cohomology. Lemma 1.1 supplies exactly the intersection-acyclicity hypothesis of Čech cohomology, Theorem 3.2. This comparison is proved for module sheaves on arbitrary ringed spaces, so it applies also to the singular and nonreduced complex spaces considered here. Theorem 2.1 of the same lesson permits the ordered Čech complex. For a cover by \(m\) open sets this complex has no terms in degrees \(q\geq m\), because a term in degree \(q\) requires \(q+1\) distinct indices. This gives the vanishing range in Corollary 1.2. The comparison commutes with refinement: the restriction maps on the injective-resolution double complex fix the augmented global-section complex. Thus the nested-cover comparison in Theorem 2.1 induces the identity on sheaf cohomology, as needed for the compact-restriction argument.

Corollary 1.2. Every covering of a complex space by Stein pieces is a Leray covering for every coherent sheaf: its Čech complex computes the sheaf cohomology. In particular, if \(X\) is covered by \(m\) Stein pieces, then \(H^q(X,\mathcal S)=0\) for every coherent \(\mathcal S\) and every \(q\geq m\).

Proof. By Lemma 1.1 and Stacks, Tag 01ET. With \(m\) opens, the alternating Čech complex has no cochains in degrees \(q\geq m\). \(\square\)

For instance, complex projective space \(\mathbf P^n\) is covered by the \(n+1\) Stein open sets \(\{T_j\neq0\}\cong\mathbf C^n\), so \(H^q(\mathbf P^n,\mathcal S)=0\) for \(q>n\) and every coherent analytic sheaf \(\mathcal S\); the same holds on every closed complex subspace of \(\mathbf P^n\).

2. The finiteness theorem

Theorem 2.1 (Cartan–Serre). Let \(X\) be a compact complex space and \(\mathcal S\) a coherent \(\mathcal O_X\)-module. Then \(H^q(X,\mathcal S)\) is finite-dimensional for every \(q\geq0\).

Proof. For each point of \(X\) choose a local model \(A\subset\Omega\subset\mathbf C^N\) which is a distinguished patch for \(\mathcal S\) (Stein domains in complex space, Proposition 4.3), and concentric balls \(V'\Subset V\Subset\Omega\) about the point. By compactness, finitely many of the Stein pieces \(U'_\alpha=A_\alpha\cap V'_\alpha\) cover \(X\); put \(U_\alpha=A_\alpha\cap V_\alpha\). The coverings \(\mathcal U=(U_\alpha)\) and \(\mathcal U'=(U'_\alpha)\) are Leray coverings by Corollary 1.2, so both Čech complexes compute \(H^\bullet(X,\mathcal S)\), and the restriction map \(\rho:C^\bullet(\mathcal U,\mathcal S)\to C^\bullet(\mathcal U',\mathcal S)\) induces the identity of \(H^\bullet(X,\mathcal S)\), in particular a surjection.

Each closure \(\overline{U'_\alpha}\subset A_\alpha\cap\overline{V'_\alpha}\) is compact and contained in \(U_\alpha\), so the closure of every finite intersection \(U'_{\alpha_0\ldots\alpha_k}\) is a compact subset of \(U_{\alpha_0\ldots\alpha_k}\). The Čech cochain spaces are finite products of the Fréchet spaces \(\mathcal S(U_{\alpha_0\ldots\alpha_k})\), and by Fréchet spaces of sections, Proposition 5.2(3) each restriction \(\mathcal S(U_{\alpha_0\ldots\alpha_k})\to\mathcal S(U'_{\alpha_0\ldots\alpha_k})\) is compact. So \(\rho\) is compact in every degree, and the finiteness criterion Fréchet spaces of sections, Theorem 2.2 shows that \(H^q(X,\mathcal S)\) is finite-dimensional, and Hausdorff, for every \(q\). \(\square\)

Reference: [Cartan–Serre 1953] states and proves this theorem; the proof through Schwartz's theorem is arranged as in [Demailly].

Corollary 2.2. For every coherent sheaf \(\mathcal S\) on a compact complex space, the space of global sections \(\mathcal S(X)\) is finite-dimensional, and if \(X\) is covered by \(m\) Stein pieces, only the degrees \(q<m\) can be nonzero.

Proof. Theorem 2.1 with \(q=0\), and Corollary 1.2. \(\square\)

The exact inputs to Theorem 2.1. The proof keeps two uses of Stein geometry separate. Stein domains, Proposition 4.3 gives the local distinguished patches from which the nested balls are chosen. To handle intersections of different patches, Lemma 1.1 instead uses its already-proved closed product embedding. The product is Stein by PSH Proposition 2.2. Coherence and equality of cohomology under the closed embedding are the explicit closed-model comparison, followed by global Theorem B. The comparison is local for coherence, so it applies to the closed embedding into the product by using ambient coordinate charts and the coherent graph ideals of Lemma 1.1. The cohomology argument uses exact closed pushforward and identical section complexes. It does not assume that the complex space is smooth or reduced, or that one global finite free resolution exists on it.

The ordered-complex and refinement-compatible comparison were identified after Lemma 1.1. Their exact providers are Čech Theorem 2.1 and Theorem 3.2. In degree \(q\), only finitely many increasing tuples from the finite cover occur. Each smaller intersection has compact closure inside the larger one: its closure is contained in the intersection of the finitely many compact closures \(\overline{U'_\alpha}\), all lying in their corresponding \(U_\alpha\). Hence the section restriction is compact by Fréchet Proposition 5.2(3), and a finite product of these maps is compact. This also covers degree zero; an empty intersection contributes the zero section space.

Now apply the compact cohomology criterion, Theorem 2.2 to the two Fréchet complexes and this compact comparison. The induced cohomology map is the identity under the natural Leray identifications, hence is onto. The criterion gives finite dimension and Hausdorffness; neither is a prerequisite for this step. If \(X\) is empty all its groups are zero directly. Otherwise a finite cover of cardinality \(m\) also gives the stated vanishing for \(q\geq m\), since its ordered complex then has no terms. Compactness of \(X\), not a uniform bound on resolutions on arbitrary noncompact spaces, is what makes this a finite comparison.

Exact compact-finiteness input comparison: GPT-6 Astra (OpenAI), Ultra, 13 October 2026; author self-check, CC0. The product-embedding proof and Cartan–Serre theorem are preserved, not replaced or weakened.

3. The analytic results used for the comparison theorems

The proof of Serre's comparison theorems and Chow's theorem in Complex analytic spaces and analytification and Serre's comparison theorems and Chow's theorem uses the following analytic results, all proved in this course in the generality stated there.

Result Where it is proved
Weierstrass preparation and division; \(\mathcal O_n\) Noetherian with completion the formal power series ring Complex analytic spaces and analytification, Section 2
\(\mathcal O_n\) is a unique factorization domain The local ring of holomorphic germs, Theorem 3.1
Oka's coherence theorem Coherent sheaves and Oka's coherence theorem, Theorem 2.1
The local analytic Nullstellensatz Analytic germs, local parametrization and the Nullstellensatz, Theorem 5.1
Finite projections of irreducible germs; local dimension equals the Krull dimension of the local ring Analytic germs, Theorems 4.1 and 5.4
Cartan's coherence of vanishing ideals of analytic subsets, including singular ones Cartan's coherence theorem and complex spaces, Theorem 1.1
Vanishing of coherent cohomology in positive degrees on complex manifolds with a smooth strictly plurisubharmonic exhaustion Theorems A and B on Stein manifolds, Theorem 4.1
Finite-dimensionality of coherent cohomology on compact complex analytic spaces Theorem 2.1 above

In particular, the standard affine cover of \(\mathbf P^n\) and all intersections of its members are Stein (Theorems A and B on Stein manifolds, Corollary 4.2), so the Čech complex of this cover computes the analytic cohomology of every coherent analytic sheaf on \(\mathbf P^n\).

4. Exercises

Exercise 4.1. Compute \(H^q(\mathbf P^1,\mathcal O(d))\) for the analytic line bundles \(\mathcal O(d)\), using the cover by two copies of \(\mathbf C\), and check that the dimensions are finite, as Theorem 2.1 asserts.

Solution. The cover \(\{T_0\neq0\},\{T_1\neq0\}\) consists of two Stein sets with Stein intersection \(\mathbf C^*\), so it is a Leray cover. With the coordinate \(z=T_1/T_0\) and Laurent expansions on \(\mathbf C^*\), the Čech complex is \(\mathcal O(\mathbf C)\oplus\mathcal O(\mathbf C)\to\mathcal O(\mathbf C^*)\), where the first summand contributes the exponents \(a\geq0\) and the second, after the transition factor \(z^d\), the exponents \(a\leq d\). Hence \(h^0=\max(d+1,0)\) (exponents \(0\leq a\leq d\)) and \(h^1=\max(-d-1,0)\) (exponents \(d<a<0\)), and \(h^q=0\) for \(q\geq2\). This agrees with the computation in the lesson on Serre's comparison theorems.

Exercise 4.2. Show that Theorem 2.1 fails for noncompact spaces, even for the structure sheaf of a Stein manifold in degree zero.

Solution. \(\mathcal O(\mathbf C)\) contains all polynomials, an infinite-dimensional space. The proof breaks down because the comparison of two coverings, one relatively compact in the other, is only available for finitely many relatively compact pieces covering a compact space.

Exercise 4.3. Let \(X\) be a compact complex space covered by \(m\) Stein pieces. Show that \(\chi(X,\mathcal S)=\sum_q(-1)^q\dim H^q(X,\mathcal S)\) is defined for every coherent \(\mathcal S\) and is additive in short exact sequences.

Solution. By Theorem 2.1 and Corollary 1.2, only the finitely many degrees \(q<m\) contribute, each finite-dimensional. A short exact sequence of coherent sheaves gives a long exact cohomology sequence of finite-dimensional spaces of finite length, and the alternating sum of dimensions in a finite exact sequence vanishes.

References