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Abel’s theorem for divisors on an arbitrary compact complex manifold. (English) Zbl 0943.32004

Let \(M\) be a compact complex manifold of dimension \(n\), \(\text{Div}(M)\) be the abelian group of divisors on \(M\) and \({\mathcal M}(M)\) be the field of meromorphic functions on \(M\).
The main purpose of the present paper is to compute directly the composite injection \[ \begin{aligned} Cl^0(M)\hookrightarrow\text{Pic}^0(M) & \cong H^1(M,O_M)/H^1(M,\mathbb{Z})\\ & \cong H^{n,n-1}_{\overline\partial}(M)^*/H_{2n-1}(M,\mathbb{Z})\end{aligned} \] using the Čech cohomology and to show that the map is induced by \[ \text{Div}^0(M)\ni D\mapsto(H_{\overline\partial}^{n,n-1}(M)\ni[\omega]\mapsto\int_Q\omega\in \mathbb{C})\bmod H_{2n-1}(M,\mathbb{Z}), \] where \(H_{\overline\partial}^{n,n-1}(M)\) denotes the Dolbeault cohomology, \(Q\) is an integral \((2n-1)\)-chain on \(M\) with \(\partial Q=D\) and \[ \text{Div}^0(M):=\{D\in\text{Div}(M)\mid\text{ the class of }D\text{ in } H_{2n-2}(M,\mathbb{Z})\text{ is }0\}, \]
\[ Cl^0(M):=\text{Div}^0(M)/\{(F)\in\text{Div}(M)\mid F\in{\mathcal M}(M)^\times\}. \] (If \(M\) is Kähler, one can deduce this fact from Kodaira’s formula concerning a multiplicative function \(F\) (which is a kind of multivalued meromorphic function on \(M)\) such that \(D=(F)\) based on the theory of harmonic integrals.) In our proof, it appears naturally a \(C^\infty\) solution of the multiplicative Cousin problem with the data \(D\) and then we use a logarithmic residue formula for 1-forms and de Rham’s theory applied to the open submanifold \(M-\text{Supp} D\).

MSC:

32A20 Meromorphic functions of several complex variables
32C30 Integration on analytic sets and spaces, currents
58A12 de Rham theory in global analysis
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