290 M. GRINBERG AND R. MACPHERSON, EULER CHARACTERISTICS
5.5. Morse local systems
While the construction of the question mark in Section 5.3 is a difficult problem,
Fary functors can be studied microlocally. The key definition is that of a Morse
local system.
Let F be a Fary functor on (X, S). Recall that we denote by As c T* X the
co normal variety to S , and by A~ the set of generic co normals (the smooth part of
As). For each integer i, we define a Morse local system Mi (F) on A~.
Let S be a stratum, p a point in S, and ~ a generic conormal at p. If S is
open in X (and therefore~ is zero), then the stalk Mi (F)t, of Mi (F) at ~ is equal
to Hi(p; F). If S is not open, we let Mi (F)t, = Hi(Y, Z; F), where (Y, Z) is the
following pair. Fix a Riemannian metric on X , and think of ~ as a normal vector to
S at p. Let q be a point small distance E away from p, chosen to lie on the geodesic
defined by p and~· Then we set:
Y = { x E XIE - 8 ::; dist(x, q) ::; E + 8},
Z = {x EX Jdist(x,q) = E - 8},
where 0 « 8 « E « l.
Exercise 5.5.1. Show that p is the only stratified critical point of the function
dist(*,q): X -t IR, lying inside Y.
Basically, the Morse groups M*(F)t, measure the change in homology (withcoefficients in F) that occurs near p for any Morse function g : X -t IR with dμg =
-~. We chose the distance-to-q function in the definition only for convenience.Exercise 5.5.2. Use the monodromy construction of Section 5.2 to show that the
groups Mi(F)f., for different~, fit together to form a local system on A~.
Note that the fibers Mi (F)t,, for ~ in different connected components of A~,
need not be isomorphic.
Exercise 5.5.3. Assume X is oriented. Let f = xF. Show that the multiplicity
of the characteristic cycle Ch(!) at a covector ~ is equal to l:i (-l)i rank Mi (F)t,.
Exercise 5.5.4. Compute the Morse local systems of F 1 and F2 in Example 5.1.2.
5.6. Perverse sheaves
Suppose X is a complex algebraic manifold, and S is a complex algebraic stratifi-
cation. This is an especially nice situation. In particular, all strata have even real
dimension, and all the lo ci Ai are connected.
A Fary functor F on X is called a perverse sheaf, if for each stratum S E S ,
the lo cal system Mi (F) is zero on Ai, unless i =dime X - dime S.
Exercise 5.6.1. Show that both Fary functors in Example 5.1.2 are perverse
sheaves.
It is difficult to motivate this definition in the little time we have left. Let us
just mention that perverse sheaves on (X, S) form an abelian, artinian subcategory
of the category of Fary functors, and that it is closed under Verdier duality. Here
are some of the sources of perverse sheaves in mathematics: