## 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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