LECTURE 1
Moduli Spaces of Stable Maps
1. Example: Quantum cohomology of complex projective spaces
In quantum cohomology theory it is convenient to think of cup-product operation
on cohomology in Poincare-dual terms of intersection of cycles. In these terms
the fundamental cycle represents the unit element 1 in H(<CPn), a projective
hyperplane represents the generator p E H^2 (<CPn), intersection of two hyperplanes
represents the generator p^2 E H^4 (<CPn), and so on. Finally, the intersection point
of n generic hyperplanes corresponds to pn E H^2 n(<CPn) and one more intersection
with pis empty so that H(<CPn) = Q[p]/(pn+l) 1.
Exercise. Check that the Poincare intersection pairing(·,·) is given by the formula
l
cf>/\ 'lj; = -2^1 · f cf>(p)'lj;(p) dp n+l ·
[<CPn] 7ri pThe structural constants (a U b, c) of cup-product count the number of inter-
sections of the cycles a, b, c in general position (taken with signs prescribed by
orientations).
The structural constants (a ob, c) of the quantum cup-product count the number
of holomorphic spheres <CP^1 ~ cpn passing by the points 0, 1, 00 through the
generic cycles a, b, c. In our example they are given by the formulas
{q^0 if k + l + m = n }
(pkop^1 ,pm)= q^1 if k+l+m=2n+l.
0 otherwiseThe first row corresponds to degree 0 holomorphic spheres which are simply points
in the intersection of the three cycles. The second row corresponds to straight lines:
all lines connecting projective subspaces pk and pm form a projective subspace of
dimension n - k + n - m + 1 = l which meets the subspace p^1 of codimension l at
one point. The degree 1 of straight lines in <CPn is indicated by the exponent in
q^1. In general the monomial qd stands for contributions of degree d spheres.
(^1) We will always assume that coefficient ring is IQ unless another choice is specified explicitly.
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