1.1. INTRODUCTION 5

exists and defines the local index form Ω(Egeom). Here Egeom is our notation for

a geometric family which is just a shorthand for the collection of data needed to

define the Bismut superconnection (see Definition 2.2.2). The following equality is

the local index theorem for families

[Ω(Egeom)] = dR(ch(index(Egeom))) ,

where index(Egeom) is our notation for the index of the family of Dirac operators

associated with Egeom.

1.1.2.6. The local index form can be determined in a similar way as in 1.1.2.2. First

assume that the vertical tangent bundle T vπ has a spin structure. Then we can

write the family of Dirac bundles as twisted spinor bundle V = S(T vπ) ⊗ W. In

this case we have

Ω(Egeom) =

E/B

ˆ

A

(∇T

v

π)ch(∇W

) ,

where the connection ∇T

v

π is induced by the data of the geometric family (see [7],

Ch. 10). In the general case the spin assumption is satisfied locally on E so that

we can obtain the integrand for the local index form by the same formula.

1.1.2.7. If index(Egeom) = 0, then the form Ω(Egeom) is exact. The main idea of

secondary index theory is to find a reason why the index is trivial and to employ

this reason in order to define a form α such that dα = Ω(Egeom).

In the present paper the reason for index(Egeom) = 0 is that Egeom admits a

taming (see 1.1.1.9). Let Et be a geometric family with a choice of a taming. Using

the taming Q we further modify the superconnection to

At(Et) := tD +

∇u

+

1

4t

c(T ) + tχ(t)Q ,

where χ is a cut-off function which vanishes for t ≤ 1 and is equal to one for t ≥ 2.

This modification has the following effects. For small t the modified superconnection

is the Bismut superconnection so that we can use the knowledge about the small

t-behavior. For large t the zero-form part is the invertible operator t(D + Q). This

has the effect that Trse−At

2(Et)

and the integrand Trs

∂At(Et)

∂t

e−At

2(Et)

in (1.2) vanish

exponentially for large t.

We can define the η-form

η(Et) :=

(2πi)− deg /2

∞

0

Trs

∂Au(Et)

∂u

e−Au(Et)du

2

.

Then we have

dη(Et) = Ω(Egeom) .

Note that the eta form depends on the taming. In fact, if Et is given by a second

taming, then η(Et)−η(Et) is closed and represents the Chern character of an element

of

K1(B)

which measures the difference of the two tamings (see Corollary 2.2.19).

The picture described above for the even-dimensional case has a analogous

odd-dimensional counterpart.

1.1.3. Absolute and relative secondary invariants.