LECTURE 4. SINGULARITY THEORY 257manifolds. Compare these results with properties of complex "oscillating" integrals
with constant phase functions.The actual motivation of the generalized mirror conjecture comes from sup-
porting examples based on toric geometry. Here is the simplest one.Example: the mirror of <CP^1 • The integral
I = r e(Y1 +Y2)/ll.
lrc{Y1Y 2 =q}
satisfies the Bessel differential equation
{)
(liq fJq)
2
I= ql
and therefore (Yq, f q, Wq) where Yq is given by the equation y 1 Y2 = q in Y = C.^2 , the
function fq is the restriction to Yq off = y 1 + y2, and wq is the relative "volume"
form (dy1 /\ dy2)/d(Y1Y2) on Yq can be taken on the role of the mirror partner of
X = <CP^1. In greater detail, let Ii > 0, q -/=-0. The function f q in the coordinate
Y1 -/=- 0 on Yq reads fq = YI + q/y1 and has two critical points y 1 = ±q^112 with the
critical values ±2q^112. On the line of values of f q pick two paths starting from the
critical values and going to infinity toward the direction Re fq -+ -oo. Each such a
path has two preimages in Yq which glue up to a non-compact cycle when oriented
oppositely. The integral I over each of the two cycles (denote them r ±) converges.
As a function of q it satisfies the Bessel differential equation. Indeed, in logarithmic
coordinates Ti = ln Yi, t = ln q the integral takes on the form
( (eTi+eT2)/n dT1 /\ dT21t±c{T 1 +T 2 =t} e d(T1 + T2)'
and the double differentiation li^282 /fJT 1 fJT 2 yields the amplitude factor eT^1 +T^2 = q.
The variety L generated by the family f q is described by the relation p^2 = q in
QH*(<CP^1 ). The potential u = f p dlnq of the action 1-form coincides with the
the critical value function ±2q^112 = 2p. Since the Hessian D.(p) of fq = eT^1 + qe-T^1
at the critical points equals 2p, the residue pairing defines the Frobenius structure
on <C[L] identical to the Poincare pairing (27ri)-^1 f dp </n/J/(p^2 - q).
Exercises. (a) Using holomorphic version of the Morse lemma show that all critical
points of the real part of a holomorphic Morse function in m variables have the
same Morse index m. Deduce that under some transversality assumptions about a
holomorphic function f : Y -+ <C at infinity the rank of the relative homology group
Hm (Y, Ref -+ -oo) equals the total multiplicity of critical points. Generalize to
higher dimensions the construction of the cycles r ± from the above example. Show
that in the example the cycles r ± form a basis in the group H 1 (Yq, Re fq-+ -oo).
Find the place for the compact cycle I Yi I = 1 in this group.
(b) Prove that (Yq, fq,wq) withYq: YI···Yn+I = q, fq = ( YI+···+ Yn+I )I Yq, Wq = dy1 ( /\ ··· /\ dyn+I )
d YI···Yn+I
is the mirror partner of X = c.pn in the same sense as in the case n = 1 studied
in the example.