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Algebraic Geometry

By B.H. Gross, B. Mazur

Publication via Gross, B.H.

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This book contributes further generalizations to the list. But, the main advantage of stratified Morse theory is that, at least for complex varieties, it provides a unified approach through which a wide variety of generalizations can be proved and understood. What follows is a nonhistorical account. Original references to the literature for specific results are given immediately after their statements in the main portion of the text. 2. Generalizations Involving Varieties which May be Singular or May Fail to be Closed One of the most dramatic generalizations is that the LHT holds for q uasiprojective varieties and the LHT* holds for singular varieties, both without modifying the statements: Theorem.

Perhaps the first attempt at an abstract theory of stratifications appears in Whitney's concept of a "complifold", or complex of manifolds [W4] (1947). However, we will concentrate on the history of stratification theory during the period between 1950 and 1970, when complete proofs of the isotopy lemmas appeared. Although stratification theory developed together with the theory of singularities of smooth mappings, it quickly became an important tool with a broad range of applications which extends well beyond the study of singularities of mappings (see, for example, [L03] (1959), [Sc] (1965), [W2] (1965), [Fe1] (1965), [Fe2] (1966), [Z3] (1971), [MP1] (1974)).

So that (4) 1 0 1tA = I for each stratum A of Z. Any vectorfield V on lRn has a controlled lift to a vectorfield W on Z. This means that W is tangent to the strata of Z, and whenever A < B are strata we have (1) (1tA)*(WIBnTA)=WIA. , W is tangent to surfaces of constant PA)· (3) I*(W)= V. It turns out that the integral curves of such a controlled vectorfield W fit together (stratum by stratum) to give a continuous one parameter family of stratum preserving homeomorphisms of Z which commute with f Furthermore, commuting vectorfields Vi> Vz , ...

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