分段正则映射

摘要：

设V，W是实代数簇（也就是说，直到同构，实代数集），X是V的一个子集。一个从X到W的映射f称为正则的，如果它可以扩展到定义在V的某个包含X的Zebrski局部闭子簇上的正则映射。此外，如果存在V的一个分层，使得f对X与每个分层的交集的限制是正则映射，则这样的映射被称为分段正则映射。通过V的分层，我们意味着一个有限的收集成对不相交Zagliki局部闭子簇，其联合等于V.假设子集X是紧的，我们证明了每个连续映射从X到格拉斯曼簇或单位球可以近似分段正则映射。作为应用，我们得到了拓扑向量丛代数化定理的一个变形。如果簇V是紧的且非奇异的，我们证明了从V到单位球面的每个连续映射同伦于类C^k的分段正则映射，其中k是任意的非负整数.

英文摘要：

Let V, W be real algebraic varieties (that is, up to isomorphism, real algebraic sets), and let X be a subset of V. A map f from X into W is said to be regular if it can be extended to a regular map defined on some Zariski locally closed subvariety of V that contains X. Furthermore, such a map is said to be piecewise-regular if there exists a stratification of V such that the restriction of f to the intersection of X with each stratum is a regular map. By a stratification of V we mean a finite collection of pairwise disjoint Zariski locally closed subvarieties whose union is equal to V. Assuming that the subset X is compact, we prove that every continuous map from X into a Grassmann variety or a unit sphere can be approximated by piecewise-regular maps. As an application, we obtain a variant of the algebraization theorem for topological vector bundles. If the variety V is compact and nonsingular, we prove that each continuous map from V into a unit sphere is homotopic to a piecewise-regular map of class C^k, where k is an arbitrary nonnegative integer.

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