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Linking numbers and boundaries of varieties. (English) Zbl 1013.32007
In this paper boundaries of positive holomorphic chains in $$\mathbb C^n$$ are being characterized by non-negativity of their linking numbers with algebraic subvarieties of complementary dimension. More precisely, the following is proved.
Let $$M$$ be a compact oriented submanifold of $${\mathbb C}^n$$ of odd real dimension $$k$$, $$3\leq k\widehat A \leq 2n-3$$. Then $$M$$ satisfies the linking condition $$\text{lk}(M,A)\geq 0$$, for all algebraic subvarieties $$A$$ of $${\mathbb C}^n$$ disjoint from $$M$$ of complex dimension $$n-(k+1)/2$$, where $$\text{lk}$$ denotes the linking number, if and only if $$M$$ is maximally complex and there exists a (unique) positive holomorphic $$k$$-chain $$T$$ of dimension $$(k+1)/2$$ in $${\mathbb C}^n - M$$ of finite mass and bounded support with $$[M]=\partial[T]$$. Moreover, for all $$x\in {\mathbb C}^n - M$$, $$x\in\text{supp}(T)$$ if and only if $$\text{lk}(M,A)>0$$, for all algebraic subvarieties $$A$$ of $${\mathbb C}^n$$ disjoint from $$M$$ of complex dimension $$n-(k+1)/2$$ such that $$x\in A$$.
The proof uses the result of F. R. Harvey and H. B. Lawson jun. [Ann. Math. (2) 102, 223-290 (1975; Zbl 0317.32017)] that $$M$$ bounds a holomorphic chain if and only if it is maximally complex and is organized as follows. First the corresponding theorem for curves in $$\mathbb C^n$$ is proved and from this the theorem with $$k=n=3$$ is deduced. The case $$k=3$$ and $$n$$ arbitrary is then deduced using projections and finally the general result is proved by slicing and an inductive procedure.

##### MSC:
 32C30 Integration on analytic sets and spaces, currents 32S70 Other operations on complex singularities 57R99 Differential topology
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