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On the homotopy type of complete intersections. (English) Zbl 1072.14063
Let \({\mathbf d}=(d_1,\dots ,d_k)\) be an unordered \(k\)-tuple, \(X_n({\mathbf d}) \subset {\mathbb C}\mathbb P^{n+k}\) a complete intersection i.e. the transversal intersection of \(k\) hypersurfaces defined by vanishing of homogeneous polynomials of degrees \(d_1,\dots ,d_k.\) Let \(d=d_1d_2\cdots d_k\) be the total degree of \({\mathbf d}.\) Natural questions that arise are the classification problems for \(X_n({\mathbf d}).\) The authors study the homotopy classification problem i.e. when two complete intersections have the same homotopy type. For \(d\) which does not have small divisors and \(n\) odd, A. Libgober and J. Wood [Topology 21, 469–482 (1982; Zbl 0504.57015)] and F. Fang for \(n\) even [Comment. Math. Helv. 72, No. 3, 466–480 (1997; Zbl 0896.14028)] have proved that homotopy type of \(X_n({\mathbf d})\) is determined by \(n, d,\) Euler characteristic and signature of \(X_n({\mathbf d}).\) The authors solve this classification problem for \(n\) large relative to \(d.\) For a natural number \(k\) let \({\nu}_p(k)\) be the highest exponent of \(p\) that divides \(k.\) If \({\mathbf d}=(d_1,\dots,d_k)\) define its \(p\)-localization to be the sequence obtained from \((p^{{\nu}_p(d_1)},\dots ,p^{{\nu}_p(d_k)})\) by deleting entries with \({\nu}_p(d_i)=0.\) One calls \({\mathbf d}\) to be equivalent to \({\mathbf {d'}}\) if for all \(p\) their \(p\)-localizations are the same. The authors prove the following theorems.
Theorem 1.1. If \({\mathbf d}\) and \({\mathbf {d'}}\) are equivalent, then for any \(n>2\) the complete intersections \(X_n({\mathbf d})\) and \(X_n({\mathbf {d'}})\) are homotopy equivalent provided their Euler characteristics and signatures agree.
Theorem 1.2. If \({\mathbf d}\) and \({\mathbf {d'}}\) are nonequivalent multidegrees, there is a positive integer \(N({\mathbf d},{\mathbf {d'}})\) such that if \(n\geq N({\mathbf d},{\mathbf {d'}})\) then the complete intersections \(X_n({\mathbf d})\) and \(X_n({\mathbf {d'}})\) are not homotopy equivalent.
The authors derive from these theorems two interesting corollaries one of which says when (under some additional hypotheses on \(n, {\mathbf d}\) and \({\mathbf {d'}}\)) the complete intersections \(X_n({\mathbf d})\) and \(X_n({\mathbf {d'}})\) are homeomorphic.

14M10 Complete intersections
55P15 Classification of homotopy type
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