Relative Morse Theory

The reader who is interested only in Morse theory for singular spaces or for nonproper Morse function may skip this chapter. We will consider the Morse theory of a composition $$ X\xrightarrow{R}Z\xrightarrow{f}\mathbb{R} $$ which will eventually be used (in Part II, Sects. 5.1 and 5.1*, with Z = ℂℙn) to prove a conjecture of Deligne [Dl] concerning Lefschetz hyperplane theorems for a variety X and an algebraic map π: X → ℂℙn. We will approximate the function f by a Morse function, although the composition fπ: X → ℝ is not Morse (or even Morse-Bott) in any reasonable sense. All attempts to prove Deligne’s conjecture by approximating fπ by a Morse function seem to end in failure because one loses curvature estimates on the Morse index of fπ. Instead, we are forced to “relativize” the Morse theory of f. Our main result is stated in Sect. 9.8.