The example we study first appeared in the physics literature in
[10]. We will describe this example precisely in what follows, but
first let us emphasize the basic idea. We start with a smooth quintic
in 
and follow a path in its complex structure moduli space leading
us to a conifold with 16 singular points. Each of these singular points can
be described locally as a cone over an 
base. We resolve such
singularities by cutting out
a neighborhood of the singular point and gluing in
a smooth space (of real dimension 6) whose boundary
agrees with that of the extracted set,
namely 
.
For the singular points of a conifold this can
be done in two ways: i) glue in 
or ii) glue in 
.
The former is a deformation of the conifold back into the
quintic moduli space, by giving positive volume
to the shrunken three-cycles. The latter is a small resolution of the conifold
by
giving positive area to the 
's. Effectively, the small resolution
replaces
previous 
's with 
's and thereby changes the Hodge numbers, and hence
topology, of the Calabi--Yau space. (In fact, there are generally two distinct
ways of performing the small resolution that differ by a flop. This played
a key role in [1,2] but is not of central importance here.)
Our goal is to understand how type II string theory behaves as we attempt
to pass from the smooth quintic, through the conifold, and on to its
topologically distinct small resolution. To do so, we first recast the present
discussion into a more concrete form.
Consider the set of quintics in 
which contain a fixed
, say the one with 
. We do this because,
as we shall see, the 16 singular points referred to above can be made
to all reside on this 
. The defining equation of
such a Calabi--Yau space must not contain any of the 21 monomials
, ..., 
which involve only 
, 
and 
;
there are therefore 105 parameters in the defining equation.
On the other hand, the number of redundancies has decreased,
since we are only free to use the subgroup of 
which fixes the 
, i.e., matrices 
for which the
coefficients 
vanish when j=3, 4 and k=0, 1, 2.
That subgroup has dimension 19, so the
total number of effective parameters is 86, a set of codimension 15
in the original 101-dimensional space 
.
If we write the defining polynomial of such a quintic in the form
where 
and 
are polynomials of degree 4,
then by considering the partial derivatives of f with respect to
and 
it becomes apparent that the quintic must
be singular along the set 
which consists
of the sixteen points of intersection of g and h within 
.
When g and h are generic, these singularities are simply
nodes. That is, we have defined a Calabi--Yau conifold
rather than a Calabi--Yau manifold.
The vanishing cycles of these singular points are easy to locate.
To do so, consider a neighborhood of a given singular point on the
conifold that we obtain by
intersecting the quintic with a ball in 
, i.e., a 
. As discussed in
[10,11],
this intersection is a cone with base being 
.
Therefore,
if we remove 16 such small balls
from 
about the 16 singular points, then at each point
we remove a singular portion
from the Calabi--Yau whose boundary is topologically 
.
It is then possible to glue in 16 copies of 
to
obtain the smooth quintic Calabi--Yau manifold. Note that the singular
Calabi--Yau contains 
as a smooth 4-manifold
passing through the 16 singular points. When we remove the 16 balls, we
remove
16 
's from this 
. The boundary of each such 
is the 
we have glued in to desingularize the space, i.e.,
the vanishing cycles. The 
with 16 
's removed
is thus a 4-manifold-with-boundary on our Calabi--Yau manifold, and
its boundary is precisely the sum of all the vanishing cycles 
.
That is, 6 holds in the homology group.
Some relation such as 6 was to be expected from our count
of the codimension.
The singular quintics can alternatively be given a small resolution by
blowing up the 
contained within them. We blow up the locus
in 
defined by 
, which can be modeled inside
as the set where 
, 
being homogeneous coordinates on 
. The original Calabi--Yau
conifold is blown up to a Calabi--Yau manifold defined by
Topologically, this small resolution process glues in a copy
of 
along each boundary 
. There is a new 
class on the resolved space, which measures the area of the new
's which were added. (All have the same area.)
