# Local family index theorem, but with Chern class?

Let $\pi:X\to B$ be a proper submersion with spin fibers, and $E\to X$ a Hermitian vector bundle with a unitary connection $\nabla$. Then the local family index theorem for spin Dirac operator twisted by $E$ states that $$d\widetilde{\eta}=\int_{X/B}\widehat{A}(\nabla^{T^VX})\wedge ch(\nabla)-ch(\nabla^{ker(D^E)}),$$ assuming the kernel bundle exists.

I would like to know if there is an analogous statement for Chern classes? That is, replacing the Chern character by Chern classes. And also Chern classes with other Dirac operators, for example the signature operators and de Rham operator (with the corresponding conditions on the fibers).

Thank you very much in advance.

• May you let me know which Bismut paper? – GRR Jul 18 '17 at 15:59
• The map $\alpha\mapsto\int_{X/B}\hat A(\nabla^{T^vX})\wedge\alpha$ does not preserve degrees, which makes a positive answer implausible. From the Chern character form, one can extract the Chern class forms $c_k(\nabla^{\ker(D^E)})$ one by one, but each of them will typically depend on several Chern classes of $E$. If you take the fibrewise Euler operator, the corresponding operation is Becker-Gottlieb transfer $\alpha\mapsto\int_{X/B}e(\nabla^{T^vX})\wedge\alpha$. It preserves degree, but does respect the product structure as far as I know. – Sebastian Goette Jul 19 '17 at 5:57
• The only situation I know where something similar works is in Bismut and Lott's paper. Here, the Becker-Gottlieb transfer is applied to the Kamber-Tondeur classes one by one, independent of the others. – Sebastian Goette Jul 19 '17 at 6:00
• Yes, essentially I would need to use the de Rham operator. In the Bismut Lott paper you mentioned, they deal with the imaginary part of the Cheeger Chern Simons class for complex flat vector bundle, and they can decompose it according to the degree, which gives the Chern classes of flat vector bundle. Actually I am looking at the real part of the Cheeger Chern Simons class, a R/Q class. When decompose it according to the degree, we hope to get R/Z class by factoring out some coefficients. However, those classes do not seem to admit a very explicit differential form representatives. – GRR Jul 19 '17 at 10:02
• Even if it does, it is still hard to prove such an equality. That's why I am looking for the existence of a primary local FIT for Chern classes. From your answer it seems not quite possible. – GRR Jul 19 '17 at 10:04