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Topology of complete intersections. (English) Zbl 0896.14028
The author considers complete intersections in complex projective spaces. It is a classical result of R. Thom that the topology of an \(n\)-dimensional complete intersection depends only on the degrees of the homogeneous polynomials. For an unordered set \({\mathbf d}= (d_1,\dots, d_r)\) of nonnegative integers, \(X_n({\mathbf d})\) will denote an \(n\)-dimensional complete intersection defined by \(r\) homogeneous polynomials in \(n=r+1\) variables and degrees \(d_1,\dots, d_r\), respectively. The total degree of \(X_n({\mathbf d})\) is defined as \(d= d_1d_2\cdots d_r\).
The following stems from the author’s abstract: Let \(X_n({\mathbf d})\) and \(X_n({\mathbf d}')\) be two \(n\)-dimensional complete intersections with the same total degree \(d\). If \(d\) has no prime factors less than \(\frac{n+3}{2}\) with \(n\) even, then \(X_n({\mathbf d})\) and \(X_n({\mathbf d}')\) are homotopy equivalent if and only if they have same Euler characteristics and signatures. This confirms a conjecture of Libgober and Wood. Furthermore, if \(d\) has no prime factors less than \(\frac{n+3}{2}\), then \(X_n({\mathbf d})\) and \(X_n({\mathbf d}')\) are homeomorphic if and only if their Pontryagin classes and Euler characteristics agree.

14M10 Complete intersections
13C40 Linkage, complete intersections and determinantal ideals
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