As an exercise for myself I wanted to check GRR in the following situation. Consider $P:X \rightarrow B$ to be an Weierstrass elliptic fibration with a section, and $X\times_B X$ be the fiber product with $\pi_{1,2}$ be the corresponding projections to the first and second factor. Then from GRR we have,

$ch(\pi_{2*} \mathcal{O}_{X\times_B X})=\pi_{2*} \big(ch(\mathcal{O}_{X\times_B X})\cdot \pi_1^* Td (X/B)\big) = \\ \pi_{2*} \big[\pi_1^*\big(1-\frac{1}{2}P^* c_1(B)+\frac{13}{12} P^* c_1^2(B)+\sigma P^*c_1(B)-\frac{1}{2}\sigma P^* c_1^2(B)\big)\big]$.

where $\sigma$ is the section. So obviously the left hand side is 1, but I cannot show why the right hand side should be 1.

Base $B$, is a surface (can be projective space, or Hirtzebruch), and maybe I should mention that $\sigma^2=-c_1(B)\sigma$

Let's be more specific, if I'm right we can use the following commutative diagram,

$\require{AMScd} \begin{CD} X\times_B X @>{\pi_1}>> X\\ @V{\pi_2}VV @VV{P_1}V\\ X @>>{P_2}> B \end{CD}$

Then $\pi_{2*}o \pi_1^* \sim P_{2}^* o P_{1*} $. So we can use the projection formula. Now the specific questions are,

$P_{*}(\sigma)=?$

$P_{*}(1)=?$.

Naively both should be 1, but it cannot be true...