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) = dα 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.