Let $X$ be some smooth projective variety over $\mathbb{C}$ and let
$$\mathscr{H}_{q}(X):=ch(\Omega_{X}^{q})Td(X).$$
For $Y$ a certain elliptic fibration
$$\varphi:Y\to B$$
where $B$ is of arbitrary dimension, I have been computing
$$\varphi_{*}\mathscr{H}_{q}(Y)$$
where $\varphi_{*}$ is the proper pushforward. The total space $Y$ is a subvariety of a projective bundle
$$\mathbb{P}(\mathscr{O}\oplus \mathscr{L}\oplus \mathscr{L}\oplus \mathscr{L})$$
where $\mathscr{L}$ is a line bundle on $B$. I have computed that
$\varphi_{*}Td(Y)=(1-e^{-L})Td(B)$
$$\varphi_{*}\mathscr{H}_{1}(Y)=(1-e^{-L})\mathscr{H}_{1}(B)+(-4-e^{-L}+3e^{-2L}+2e^{-3L})Td(B),$$
where $L=c_1(\mathscr{L})$. What I would like is an explanation of the result of these calculations in terms of Grothendieck-Riemann-Roch as I have only recently acquainted myself with GRR. Thanks everyone.