Todd class of blow-up

Question

Let $i:X\hookrightarrow Y$ be an embedding of two non-singular projective varieties over $\mathbb{C}$. Consider the blow-up $f:Y' = Bl_XY \to Y$, and the corresponding embedding $j:E\hookrightarrow Y'$ of the exceptional divisor. Denote by $f_X: E\to X$ the restriction map. For us, the Todd class of $Y'$ is the one associated to its tangent sheaf $\mathcal{T}_{Y'}$, i.e., $Td(\mathcal{T}_{Y'}) \in A(Y')_{\mathbb{Q}}$. I would like to compute this class in terms of $f^*Td(\mathcal{T}_{Y})$ as it was done for their Chern classes (see this post). The first approach is using the tangent exact sequence, $$0 \to \mathcal{T}_{Y'} \to f^*\mathcal{T}_{Y} \to j_*Q \to 0, $$ where $Q$ is the universal quotient sheaf of the universal exact sequence $$0 \to \mathcal{O}_{E}(-1) \to f^*\mathcal{N}_{X/Y} \to Q \to 0,$$ and $\mathcal{N}_{X/Y}$ the normal sheaf on $X$. In this way, $$ Td(\mathcal{T}_{Y'}) = \frac{f^*Td(\mathcal{T}_{Y})}{Td(j_*Q)}.$$ So, we just have to understand what is $Td(j_*Q)$.

• I was attempting this computation for the case $X = point$, since in this case we have $f^*\mathcal{N}_{X/Y} = \mathcal{O}_E$. However, little experiments show me that my method may not be correct. Let me show you my method: Push-forwarding by $j_*$ the universal exact sequence we get $$Td(j_*Q) = \frac{Td(j_*\mathcal{O}_E)}{Td(j_*\mathcal{O}_E(-1))} = \frac{1}{Td(\mathcal{O}_{Y'}(E))Td(\mathcal{O}_{Y'}(-E))},$$ where the last identity follows from the structural exact sequences $0\to \mathcal{O}_{Y'}(-E) \to \mathcal{O}_{Y'} \to j_*\mathcal{O}_E \to 0$, and the fact that $\mathcal{O}_E(-1) = \mathcal{O}_E(E)$.

So, I don't know if I'm right or if I'm committing a mistake in some line. Thank u very much for your help!