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.