UDC 513.838

MATHEMATICS

V. D. GOLOVIN

DUALITY FOR COHERENT ANALYTIC SHEAVES

(Presented by Academician I. G. Petrovskii, 15 VIII 1969)

1. Let \(X\) be a complex analytic manifold, countable at infinity, of complex dimension \(n\), and let \(F\) be a coherent analytic sheaf on \(X\). Each point of the manifold \(X\) has a holomorphically complete open neighborhood \(U\), over which there is defined an epimorphism of sheaves

\[ \pi:\ O^m \to F, \]

where \(O\) is the sheaf of germs of holomorphic functions on \(X\), and \(m\) is some positive integer. Consequently, the complex vector space \(\Gamma(U;F)\) of all continuous sections of the sheaf \(F\) over \(U\) is isomorphic to the quotient space \(\Gamma(U;O^m)/\Gamma(U;R)\), where \(R\) is the kernel of the epimorphism \(\pi\). Endowing the space \(\Gamma(U;O^m)\) with the topology of compact convergence, we thereby introduce on \(\Gamma(U;F)\) a certain separated locally convex topology, which is a Fréchet topology and does not depend on the choice of the epimorphism \(\pi\) (cf. \((^1)\)).

Let \(\mathfrak U=(U_i)\) be a sufficiently fine locally finite covering of the manifold \(X\) by holomorphically complete open sets. The space \(C^k(\mathfrak U;F)\) of cochains of degree \(k\ge 0\) of the covering \(\mathfrak U\) with coefficients in \(F\) is endowed with the product topology of the spaces \(\Gamma(U_{i_0}\cap\ldots\cap U_{i_k};F)\). Further, the space \(H^k(\mathfrak U;F)\) of cohomology of the covering \(\mathfrak U\) is endowed with the quotient-space topology \(\operatorname{Ker}\delta_k/\operatorname{Im}\delta_{k-1}\), where \(\delta_k\) is the (continuous) coboundary operator. Finally, the space \(H^k(X;F)\) of Čech cohomology of the manifold \(X\) with coefficients in the sheaf \(F\) is endowed with the topology of the inductive limit of the spaces \(H^k(\mathfrak U;F)\) with respect to the filtering set of classes of pairwise equivalent coverings. By means of a diagram chase analogous to the proof of A. Weil’s de Rham theorem (see \((^2)\)), it is easy to show that the canonical mapping

\[ H^k(\mathfrak U;F) \to H^k(X;F) \]

is an isomorphism of topological vector spaces.

1. Let \(E^k\), for any integer \(k\ge 0\), be the sheaf of germs of infinitely differentiable exterior differential forms of bidegree \((0,k)\) on the manifold \(X\). Consider the complex \(\Gamma(X;E^*\otimes_O F)\) of vector spaces \(\Gamma(X;E^k\otimes_O F)\) \((k=0,1,\ldots)\), whose coboundary operator

\[ d_k'' : \Gamma(X;E^k\otimes_O F) \to \Gamma(X;E^{k+1}\otimes_O F) \]

is induced by the exterior differential \(d'':E^k\to E^{k+1}\). For each \(U\in\mathfrak U\), endow the vector space \(\Gamma(U;E^k\otimes_O F)\) with the strongest of the locally convex topologies under which the canonical bilinear mapping

\[ \Gamma(U;E^k)\times \Gamma(U;F) \to \Gamma(U;E^k\otimes_O F), \]

which assigns to sections \(\omega\) and \(\xi\) over \(U\) of the sheaves \(E^k\) and \(F\), respectively, the section \(x\mapsto \omega(x)\otimes \xi(x)\) of the sheaf \(E^k\otimes_O F\). The space \(\Gamma(X;E^k\otimes_O F)\) is endowed with the wea-

weakest of the topologies for which the restriction mappings are continuous

\[ \Gamma(X;E^k\otimes_{\mathcal O}F)\to \Gamma(U;E^k\otimes_{\mathcal O}F)\qquad (U\in\mathfrak u). \]

Finally, the cohomology space \(H^k(\Gamma(X;E^*\otimes_{\mathcal O}F))\) of the complex \(\Gamma(X;E^*\otimes_{\mathcal O}F)\) is endowed with the topology of the quotient space \(\operatorname{Ker}d''_k/\operatorname{Im}d''_{k-1}\).

Since \(E_x^k\) is a flat \(\mathcal O_x\)-module for every \(x\in X\) (see (3)), by the Dolbeault–Grothendieck lemma the sequence of sheaves

\[ 0\to F\to O_p\otimes_{\mathcal O}F\to E^1\otimes_{\mathcal O}F\to\cdots \]

is exact. Consequently, by means of a diagram chase one can show that, for every \(k\ge 0\), the topological vector spaces \(H^k(X;F)\) and \(H^k(\Gamma(X;E^*\otimes_{\mathcal O}F))\) are canonically isomorphic.

3. Let \(a\) be a continuous linear form on the space \(\Gamma(X;E^k\otimes_{\mathcal O}F)\). If \(U\) is an arbitrary open set in \(X\), then for any \(\omega\in \Gamma_c(U;E^k)\), \(\xi\in \Gamma(U;F)\) we have a section \(x\mapsto \omega(x)\otimes \xi(x)\) over \(X\) of the sheaf \(E^k\otimes_{\mathcal O}F\), continuous in \(\omega\) and \(\xi\). Fixing \(\xi\), we obtain a continuous linear form \(\omega\mapsto a(\omega\otimes \xi)\) on the space \(\Gamma_c(U;E^k)\) of sections with compact supports, endowed with the usual topology. Thereby a \(\Gamma(U;\mathcal O)\)-linear mapping \(\varphi_U:\Gamma(U;F)\to \Gamma(U;D^{n,n-k})\) is defined, where \(D^{n,n-k}\) is the sheaf of germs of currents of bidegree \((n,n-k)\) on \(X\), and

\[ \langle \omega,\varphi_U(\xi)\rangle = a(\omega\otimes \xi) \]

for \(\omega\in \Gamma_c(U;E^k)\), \(\xi\in \Gamma(U;F)\). It is obvious that, for \(V\subset U\), the mappings \(\varphi_V\) and \(\varphi_U\) satisfy the natural compatibility conditions, i.e. the family \((\varphi_U)\) determines a certain homomorphism of \(\mathcal O\)-modules \(\varphi:F\to D^{n,n-k}\). Since the form \(a\) has compact support, the homomorphism \(\varphi\) also has compact support. Thus a canonical linear mapping of vector spaces is defined

\[ \Gamma'(X;E^k\otimes_{\mathcal O}F)\to \Gamma_c(X;\operatorname{Hom}_{\mathcal O}(F,D^{n,n-k})), \]

which assigns to a continuous linear form \(a\) on \(\Gamma(X;E^k\otimes_{\mathcal O}F)\) a homomorphism \(\varphi:F\to D^{n,n-k}\) with compact support.

If \(\varphi=0\), then \(a=0\). Indeed, with the help of a suitable refinement of the identity it is easy to verify that, on each compact subset of \(X\), any global section of the sheaf \(E^k\otimes_{\mathcal O}F\) is representable as a finite sum of sections \(\omega\otimes \xi\), where \(\omega\in \Gamma_c(U;E^k)\), \(\xi\in \Gamma(U;F)\) for some open \(U\), and \(a(\omega\otimes \xi)=\langle \omega,\varphi_U(\xi)\rangle=0\). Consequently, the mapping \(a\mapsto \varphi\) of the vector space \(\Gamma'(X;E^k\otimes_{\mathcal O}F)\) into \(\Gamma_c(X;\operatorname{Hom}_{\mathcal O}(F,D^{n,n-k}))\) is injective. We show that it is also surjective. Since the homomorphism \(\varphi\) has compact support, it suffices to define the corresponding form \(a\) on sections of the form \(\omega\otimes \xi\), with \(\omega\in \Gamma_c(U;E^k)\), \(\xi\in \Gamma(U;F)\), by the formula \(a(\omega\otimes \xi)=\langle \omega,\varphi_U(\xi)\rangle\). It is obvious that the form \(a\) so defined is continuous on \(\Gamma(X;E^k\otimes_{\mathcal O}F)\), since each of the modules \(\Gamma(U;F)\) may be regarded as having a finite number of generators over the ring \(\Gamma(U;\mathcal O)\) (for which it is enough to choose \(U\) sufficiently small and holomorphically complete).

Thus the vector spaces \(\Gamma'(X;E^k\otimes_{\mathcal O}F)\) and \(\Gamma_c(X;\operatorname{Hom}_{\mathcal O}(F,D^{n,n-k}))\) are canonically isomorphic.

Consider the mapping

\[ {}^td'':\Gamma_c(X;\operatorname{Hom}_{\mathcal O}(F,D^{n,n-k})) \to \Gamma_c(X;\operatorname{Hom}_{\mathcal O}(F,D^{n,n-k})), \]

identified with the mapping conjugate to

\[ d'':\Gamma(X;E^k\otimes_{\mathcal O}F)\to \Gamma(X;E^{k+1}\otimes_{\mathcal O}F). \]

For any sections \(\omega\in \Gamma_c(U;E^k)\), \(\xi\in \Gamma(U;F)\) we have:

\[ \langle \omega,{}^td''\varphi_U(\xi)\rangle = {}^td''a(\omega\otimes \xi) = a(d''\omega\otimes \xi) = (-1)^{k+1}\langle \omega,d''\varphi_U(\xi)\rangle, \]

i.e., the mapping \(d''\) is induced by the sheaf homomorphism

\[ (-1)^{k+1}d'':\; D^{n,n-k-1}\to D^{n,n-k}. \]

4. Let \(D^{n,k}\), for \(k\geq 0\), be the sheaf of germs of currents of bidegree \((n,k)\) on the manifold \(X\). The corresponding Dolbeault–Grothendieck resolution of the sheaf \(\Omega^n\) of germs of holomorphic differential forms of degree \(n\) on \(X\) can be embedded in the commutative diagram

\[ \begin{array}{cccccc} & 0 & 0 & 0 \\ & \downarrow & \downarrow & \downarrow \\ 0\to & \Omega^n & \to D^{n,0} & \to D^{n,1} & \to \cdots \\ & \downarrow & \downarrow & \downarrow \\ 0\to & L^0 & \to L^{0,0} & \to L^{0,1} & \to \cdots \\ & \downarrow & \downarrow & \downarrow \\ 0\to & L^1 & \to L^{1,0} & \to L^{1,1} & \to \cdots \\ & \downarrow & \downarrow & \downarrow \end{array} \]

with exact rows and columns and with injective modules \(L^p, L^{p,q}\) \((p,q\geq 0)\) over the sheaf of rings \(O\). Since the sheaf \(F\) is coherent, \(\operatorname{Hom}_O(F,G)_x \approx \operatorname{Hom}_{O_x}(F_x,G_x)\) for any \(O\)-module \(G\) (see \((4)\)). On the other hand, from Malgrange’s theorem \((3)\) it follows that \(D_x^{n,k}\) is an injective \(O_x\)-module for every \(x\in X\). Consequently, there is a commutative diagram

\[ \begin{array}{cccccc} & 0 & 0 & 0 \\ & \downarrow & \downarrow & \downarrow \\ 0\to & \Gamma_c\!\left(X;\operatorname{Hom}_O(F,\Omega^n)\right) & \to \Gamma_c\!\left(X;\operatorname{Hom}_O(F,D^{n,0})\right) & \to \Gamma_c\!\left(X;\operatorname{Hom}_O(F,D^{n,1})\right) & \to \cdots \\ & \downarrow & \downarrow & \downarrow \\ 0\to & \Gamma_c\!\left(X;\operatorname{Hom}_O(F,L^0)\right) & \to \Gamma_c\!\left(X;\operatorname{Hom}_O(F,L^{0,0})\right) & \to \Gamma_c\!\left(X;\operatorname{Hom}_O(F,L^{0,1})\right) & \to \cdots \\ & \downarrow & \downarrow & \downarrow \\ 0\to & \Gamma_c\!\left(X;\operatorname{Hom}_O(F,L^1)\right) & \to \Gamma_c\!\left(X;\operatorname{Hom}_O(F,L^{1,0})\right) & \to \Gamma_c\!\left(X;\operatorname{Hom}_O(F,L^{1,1})\right) & \to \cdots \\ & \downarrow & \downarrow & \downarrow \end{array} \]

in which all rows and columns, beginning with the second, are exact. By means of a diagram chase we obtain a canonical isomorphism of the cohomology vector spaces, respectively, of the complexes \(\Gamma_c(X;\operatorname{Hom}_O(F,L^{*,*}))\) and \(\Gamma_c(X;\operatorname{Hom}_O(F,D^{n,*}))\).

Consider the covariant functor

\[ L\to \operatorname{Hom}_{O,c}(X;F,L)=\Gamma_c\!\left(X;\operatorname{Hom}_O(F,L)\right), \]

defined on the category of \(O\)-modules and taking values in the category of vector spaces. We shall denote its derived functors by \(\operatorname{Ext}_{O,c}^k(X;F,L)\) \((k=0,1,\ldots)\). Thus, it has been proved above that for every \(k\geq 0\) the vector spaces \(\operatorname{Ext}_{O,c}^k(X;F,\Omega^n)\) and \(H^k(\Gamma_c(X;\operatorname{Hom}_O(F,D^{n,*})))\) are canonically isomorphic.

We endow the space \(\Gamma_c(X;\operatorname{Hom}_O(F,D^{n,k}))\) with the strong topology with respect to \(\Gamma(X;E^{n-k}\otimes_O F)\). Then in the space \(\operatorname{Ext}_{O,c}^k(X;F,\Omega^n)\) it is natural to define the topology canonically identified with the topology of the quotient space \(\operatorname{Ker} d_k''/\operatorname{Im} d_{k-1}''\), where

\[ d_k'':\; \Gamma_c\!\left(X;\operatorname{Hom}_O(F,D^{n,k})\right) \to \Gamma_c\!\left(X;\operatorname{Hom}_O(F,D^{n,k+1})\right) \]

is the coboundary operator of the complex \(\Gamma_c(X;\operatorname{Hom}_O(F,D^{n,*}))\). We shall denote by \(\widetilde{\operatorname{Ext}}_{O,c}^k(X;F,\Omega^n)\) the associated separated locally convex space, i.e., the quotient space of \(\operatorname{Ext}_{O,c}^k(X;F,\Omega^n)\) by the closure of zero in it.

1. Since the topological vector space \(H^k(X;F)\) is canonically identifiable with the quotient space \(\operatorname{Ker} d_k''/\operatorname{Im} d_{k-1}''\), where
   \[ d_k'':\Gamma(X;E^k\otimes_{\mathcal O}F)\to \Gamma(X;E^{k+1}\otimes_{\mathcal O}F), \]
   the dual space \((H^k(X;F))'\) may be identified with the quotient space
   \[ (\operatorname{Im} d_{k-1}'')^\circ/(\operatorname{Ker} d_k'')^\circ, \]
   where the polars are taken in
   \[ \Gamma_c(X;\operatorname{Hom}_{\mathcal O}(F,D^{n,n-k})). \]
   Since
   \[ \langle d_{k-1}''\omega,\varphi\rangle = \langle \omega^t,d_{k-1}''\varphi\rangle \]
   for all \(\omega\in\Gamma(X;E^{k-1}\otimes_{\mathcal O}F)\),
   \(\varphi\in\Gamma_c(X;\operatorname{Hom}_{\mathcal O}(F,D^{n,n-k}))\), it follows from \(\langle d_{k-1}''\omega,\varphi\rangle=0\) (for every \(\omega\)) that \({}^td_{k-1}''\varphi=0\), and conversely. In other words, \((\operatorname{Im} d_{k-1}'')^\circ\) coincides with
   \[ \operatorname{Ker}\,{}^td'': \Gamma_c(X;\operatorname{Hom}_{\mathcal O}(F,D^{n,n-k})) \to \Gamma_c(X;\operatorname{Hom}_{\mathcal O}(F,D^{n,n-k+1})). \]
   Similarly, the equality
   \[ \langle\omega,{}^td_k''\varphi\rangle=0 \]
   (for every \(\varphi\)) is equivalent to the equality \(d_k''\omega=0\), i.e. \(\operatorname{Ker} d_k''\) coincides with
   \[ (d''\Gamma_c(X;\operatorname{Hom}_{\mathcal O}(F,D^{n,n-k-1})))^\circ. \]
   In this case, \((\operatorname{Ker} d_k'')^\circ\) is the closure in the weak topology of the subspace
   \[ d''\Gamma_c(X;\operatorname{Hom}_{\mathcal O}(F,D^{n,n-k-1})) \]
   in
   \[ \Gamma_c(X;\operatorname{Hom}_{\mathcal O}(F,D^{n,n-k})). \]
   Since \(\Gamma(X;E^k\otimes_{\mathcal O}F)\) is reflexive, the weak and strong closures of subspaces in
   \[ \Gamma_c(X;\operatorname{Hom}_{\mathcal O}(F,D^{n,n-k})) \]
   coincide. Thus we have proved

Theorem 1. For \(0\le k\le n\), the vector space
\[ \widetilde{\operatorname{Ext}}_{\mathcal O,c}^{\,n-k}(X;F,\Omega^n) \]
is canonically identifiable with the space dual to the topological vector space \(H^k(X;F)\).

If the mapping
\[ d_k'':\Gamma(X;E^k\otimes_{\mathcal O}F)\to \Gamma(X;E^{k+1}\otimes_{\mathcal O}F) \]
is a homomorphism, then the dual mapping
\[ {}^td_k'': \Gamma_c(X;\operatorname{Hom}_{\mathcal O}(F,D^{n,n-k-1})) \to \Gamma_c(X;\operatorname{Hom}_{\mathcal O}(F,D^{n,n-k})) \]
has closed image. In this case the space
\[ \widetilde{\operatorname{Ext}}_{\mathcal O,c}^{\,n-k}(X;F,\Omega^n) \]
is Hausdorff and, consequently, coincides with
\[ \operatorname{Ext}_{\mathcal O,c}^{\,n-k}(X;F,\Omega^n). \]
The mapping \(d_k''\) is a homomorphism if the space \(H^{k+1}(X;F)\) is finite-dimensional (see \((^5)\)). Thus the following theorem holds; for (not necessarily) projective varieties it was first proved by Grothendieck \((^6)\).

Theorem 2. If the variety \(X\) is compact, then the vector space
\[ \operatorname{Ext}_{\mathcal O}^{\,n-k}(X;F,\Omega^n) \]
is canonically identifiable with the dual of the space \(H^k(X;F)\).

If the sheaf \(F\) is locally isomorphic to the sheaf \(\mathcal O^m\), then
\[ \operatorname{Ext}_{\mathcal O,c}^{\,n-k}(X;F,\Omega^n)\approx H_c^{n-k}(X;\operatorname{Hom}_{\mathcal O}(F,\Omega^n)). \]
In particular, if \(F=\Omega^p(V)\) is the sheaf of germs of holomorphic differential forms of degree \(p\) with values in the complex vector bundle \(V\), then
\[ \operatorname{Hom}_{\mathcal O}(F,\Omega^n)\approx \Omega^{n-p}(V'), \]
where \(V'\) is the vector bundle dual to \(V\). Consequently, we have

Theorem 3. The vector space
\[ H_c^{n-k}(X;\Omega^{n-p}(V')) \]
is canonically identifiable with the dual of the topological vector space
\[ H^k(X;\Omega^p(V)). \]

In the particular case when
\[ d_k'': \Gamma(X;E^{p,k}(V))\to \Gamma(X;E^{p,k+1}(V)) \]
is a homomorphism, we obtain from this the result of J.-P. Serre \((^5)\).

Kharkov State University
named after A. M. Gorky

Received
11 VIII 1969

CITED LITERATURE

\(^1\) H. Cartan, J.-P. Serre, C. R., 237, No. 2, 128 (1953).
\(^2\) A. Weil, Comm. Math. Helv., 26, No. 2, 119 (1952).
\(^3\) B. Malgrange, Séminaire Schwartz, 4, 23/1—23/11, 25/1—25/5 (1959—1960).
\(^4\) J.-P. Serre, Ann. Math., 61, No. 2, 197 (1955).
\(^5\) J.-P. Serre, Comm. Math. Helv., 29, No. 1, 9 (1955).
\(^6\) A. Grothendieck, Séminaire Bourbaki, 9, 149/1—149/25 (1956—1957).