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. The local index form can be determined in a similar way as in First
assume that the vertical tangent bundle T 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) =
) ,
where the connection ∇T
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. 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 = Ω(Egeom).
In the present paper the reason for index(Egeom) = 0 is that Egeom admits a
taming (see 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 +
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
and the integrand Trs
in (1.2) vanish
exponentially for large t.
We can define the η-form
η(Et) :=
(2πi)− deg /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
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.
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