Nonproper Morse Functions

Abstract

It is often necessary to consider a “Morse function” f: X → ℝ which is not proper, but which can be extended to a proper function

$$ \bar{f}:Z \to \mathbb{R} $$

where Z contains X as a dense open subset. For example, Z may be a compactification of some noncompact algebraic variety X ⊂ ℂℙⁿ, and f may be a smooth function defined on the ambient ℂℙⁿ. We shall assume that it is possible to find a stratification of Z so that X ⊂ Z is a union of strata. Thus, X is obtained from Z by removing certain strata. The main theorems (Sects. 3.7, 3.10, 3.11) continue to apply to X, because we simply remove the same strata from both sides of the homeomorphisms. Since these homeomorphisms were originally proven to be decomposition preserving, it is a triviality that they induce homeomorphisms on unions of pieces in the decomposition. However, Proposition 3.2 is no longer strictly true in this context unless we also consider the effect on X _≤v of critical values v which correspond to critical points p∈Z which do not lie in X. Our main theorems also apply to these “critical points at infinity”. For all the applications which we consider, X will be an open dense subset of Z. However, the results of this chapter apply to any union of strata X ⊂ Z, and so we will not even assume that X is locally closed in Z.