# zbMATH — the first resource for mathematics

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.

##### MSC:
 14M10 Complete intersections 13C40 Linkage, complete intersections and determinantal ideals
##### MathOverflow Questions:
Complex projective manifolds are homeomorphic if homotopy equivalent
Full Text: