6 1
1.1.3.1. To any geometric family Egeom we can associate its opposite Egeom
op
by
switching orientations and gradings. In particular we have
Ω(Egeom)
op
= −Ω(Egeom) , index(Egeom)
op
= −index(Egeom) .
Let us fix x K(B). Assume that Egeom,i, i = 0, 1, are two (nonzero-
dimensional) geometric families such that index(Egeom,i) = x. Then the difference
of local index forms Ω(Egeom,1) Ω(Egeom,0) is exact. The reason is that the fiber-
wise sum Fgeom := Egeom,0
op
+ Egeom,1 has a trivial index and thus admits a taming
Ft. We therefore can write
(1.4) Ω(Egeom,1) Ω(Egeom,0) = dη(Ft) .
Note that if we change the taming, then η(Ft) changes by a closed form with
rational periods (namely by a form which represents the Chern character of some
element of K(B), see 1.1.2.7)
1.1.3.2. The class x
K∗(B),
Z/2Z, determines a rational cohomology class
ch(x)
H∗(B,
Q)
1.
If we represent x as the index of a fixed geometric family
Egeom,0, then we obtain the additional information of a de Rham representative
Ω(Egeom,0) of ch(x). If we take another representative Egeom,1, then have the sec-
ondary information η(Ft) such that (1.4) holds true.
Thus after fixing Egeom,0, we have a relative invariant defined on the set of all
geometric families with index x which takes values in the quotient
AB
∗−1
(B)/AB
∗−1
(B, d = 0, Q)
of all differential forms on B by closed forms with rational periods.
1.1.3.3. One may ask if one can turn this relative invariant into an absolute one. In
fact there is a group2 valued functor HDel(B, Q) called rational Deligne cohomology
(see Definition 4.1.1) which may capture this kind of information. Rational Deligne
cohomology comes with two natural transformations
HDel(B,

Q)
R
AB(B,

d = 0) , HDel(B,

Q)
v

H∗(B,
Q) ,
which are called the curvature and the characteristic class, such that [Ru] =
dR(v(u)) for all u HDel(B, Q). Furthermore, there is a natural transformation
AB
∗−1
(B)
a
HDel(B,

Q)
such that Ra(α) = and
0
A∗−1(B)/AB ∗−1(B,
d = 0, Q) HDel(B,

Q)
v

H∗(B,
Q) 0
is exact, where the first map is induced by a.
If we fix an element indexDel,Q(Egeom,0) HDel(B,

Q) such that
RindexDel,Q(Egeom,0)
= Ω(Egeom) ,
then we would obtain an invariant indexDel,Q(Egeom,1) HDel(B,

Q) for all geo-
metric families Egeom,1 with index(Egeom,1) = x by the prescription
indexDel,Q(Egeom,1) := indexDel(Egeom,0) + a([η(Ft)]) .
1In
the following we consider cohomology and forms as Z/2Z-graded.
2This
in fact a ring valued functor, but the ring structure is not important in the present
paper.
Previous Page Next Page