On the homotopy type of complete intersections
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, \textit{A. Libgober} and \textit{J. Wood} [Topology 21, 469--482 (1982; Zbl 0504.57015)] and \textit{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.
scientific article; zbMATH DE number 2137349