254 A. GIVENTAL, A TUTORIAL ON QUANTUM COHOMOLOGY
(c) Show that Lis given by the characteristic equation det(A^1 - LPidti) = 0
(to be satisfied for all values of the commuting coordinates dti on t he tangent space
TtB) at least if the eigen-value 1-forms D; are everywhere distinct.
(d) Identify the commutative algebra C[B][A 1 ,... , Ar] with t he algebra of func-
tions on L (at least under the same hypotheses as in ( c)).
(e) Show that the basis of delta-functions of the branches in L diagonalizes the
inner product (¢, 'l/J) in the algebra:
(¢, 'l/J)(t) = 2=
p"'ELnT; B
<P (p°') 'ljJ (p°')
.0.(p°')
where 6 is a suitable function on L. Show that in t he quantum cohomology setting
the function 6 / (dim H) represents t he cohomology class Poincare-dual to a point.
The class in H *(X x X) Poincare-dual to the diagonal defines an element in
C[LxBL]. Show that 6 is the restriction of t his element to the diagonal LC LxBL.
Compare both descriptions of 6 with t he residue formula for Poincare pairing in
Q H * ( c_pn) from the exercise in t he section l.
(f) Consider the function u on L defined as a (local) potential J LPi dt i of
the eigen-value 1-forms. Ta king into account the grading in quantum cohomology
algebras show that the restriction to L of the linear function c 1 (Tx) = L, μ iPi plays
the role of such a potential.
Now let us try to find an asymptotical representation of a fundamental solution
S to t he differential system \l liS = 0 in the form
S = w(t)(l + li<I?(t) + o(n)) exp(U /Ii)
of t he product of a formal matrix series in Ii and the exponential function of the
diagonal matrix u I Ii. Equating the terms of order n° in the equation \l liS = 0 we
obtain A^1 w = WdU which means t hat columns of W must be eigen-vectors of A^1
and the entries of the diagonal matrix U must be potentials Uo: of the eigen-value
1-forms. In the order li^1 we have w-^1 dw = [dU, <I?]. Since commutator with a
diagonal matrix has zero diagonal entries, this means that in the variation 'l'(t) of
a n (orthogonal!) eigen-basis of A^1 inner squares of the eigen-vectors may not vary.
Exercise. Proceed to higher orders in Ii in order to show that t he asymptotical
fundamental solution in the form 'l'(l + /i<J? 1 + li^2 <J? 2 + ... ) exp(U/li) exists.
Reformulating the result of our computation in terms of the algebra of functions
on L we conclude that the system \l lis = 0 has a basis of solutions with the
asymptotical expansion so:= eu(p°'(t ))/li( 6112 (p°')(l + o(li)), and that
the corresponding component (1, so:) of the vector-function J assumes the form
eu°'/ li
lo:= 11(l+o(!i)).
vb.a
This form strongly resembles stationary phase asymptotics of oscillating integrals
in singularity theory - the subject we have to discuss next.
Let n : Y --> B be a family of complex manifolds yt and f : Y --> C be a
holomorphic function. One defines the Lagrangian variety L C T * B parametrized
by critical points of functions ft := f IYt as follows. A critical point is a point in
y E Y where the differential dy f is projectable to a covector p(y) on B applied at
t = n(y). Since f t may have several crit ical points, we obtain several covectors in
