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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.

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.

Reviewer: Tobias Ekholm (Uppsala)

### MSC:

32C30 | Integration on analytic sets and spaces, currents |

32S70 | Other operations on complex singularities |

57R99 | Differential topology |