Differential Invariants of Embeddings of Manifolds in Complex Spaces

Abstract

Let V be a reduced complex space, W a complex submanifold, and let (V′, W′) be another such pair. Let f : V → V′ be a homeomorphism with f(W) ⊂ W′, such that f and f ⁻¹ are both continuously (real-) differentiable. Then f induces a component — (with multiplicity) — preserving homeomorphism f₀ from the normal cone C(V, W) to C(V′, W′), respecting the natural ℝ* actions on these cones. Moreover, though f₀ need not respect the ℂ* actions nevertheless the induced map on Borel-Moore homology f* : H*(W) → H*(W′) takes the Segre classes of the components of C(V,W) to ±those of the corresponding components of C(V′,W). In particular we recover the differential invariance of the multiplicity of W in V.

Partially supported by the National Security Agency