Blow up and relationship between tangents sheaves Let $X = \mathbb{P}^{n}$ and $Y \subset X$ a smooth subvariety of $X$.
Let us consider the blowup morphism of $X$ along of $Y$, denoted by $\pi : \widetilde{X} \longrightarrow X$ with exceptional divisor $E = \pi^{-1}(Y)$. 
1) Is there any relationship between $T_{\widetilde{X}}$ and $T_{X}$?
If $c \in \mathbb{Z}$ and $\mathcal{F}$ is a coherent sheaf in $X$, then :
2) Is it possible to write $T_{\widetilde{X}} \simeq \pi^{*}(T_{X}) \otimes \pi^{*}(\mathcal{F}) \otimes  \mathcal{O}_{\widetilde{X}}(cE)$?
3) Is there any relationship between $c$ and $\text{deg}(Y)$?
References on the subject and any help are welcome.
Thanks a lot.
 A: There's an exact sequence $$0\to \pi^*\Omega^1_X\to \Omega^1_{\widetilde{X}}\to i_*\Omega^1_{E/Y}\to 0$$ where $i:E\to \widetilde{X}$ is the inclusion of the exceptional divisor. This corresponds to the fact that pulling back differential forms gives you differential forms which are constant along the fibers. Checking that this is exact may be done from looking at stalks: stalks of pullbacks are easy to compute, and stalks of direct images along closed immersions are easy to compute, plus each $\Omega^1$ is locally free on it's appropriate space, as everything involved here is smooth.
Taking $\mathcal{Hom}(-,\mathcal{O}_{\widetilde{X}})$ of this exact sequence, we get the long exact sequence $$ 0\to \mathcal{Hom}(i_*\Omega^1_{E/Y},\mathcal{O}_{\widetilde{X}}) \to \mathcal{Hom}(\Omega^1_\widetilde{X},\mathcal{O}_{\widetilde{X}}) \to \mathcal{Hom}(\pi^*\Omega^1_X,\mathcal{O}_{\widetilde{X}}) \to \mathcal{Ext}^1(i_*\Omega^1_{E/Y},\mathcal{O}_{\widetilde{X}})\to \cdots$$
and as $\mathcal{Hom}(-,\mathcal{O}_X)$ is exact on locally free sheaves, we see that $ \mathcal{Ext}^i(\pi^*\Omega^1_X,\mathcal{O}_{\widetilde{X}})$ and $\mathcal{Ext}^i(\Omega^1_\widetilde{X},\mathcal{O}_{\widetilde{X}})$ both vanish for $i>0$. As $i_*\Omega^1_{E/Y}$ is torsion, it's sheaf hom in to $\mathcal{O}_\widetilde{X}$ vanishes, and so all that's left to do is to identify $\mathcal{Ext}^1(i_*\Omega^1_{E/Y},\mathcal{O}_\widetilde{X})$. Replacing $i_*\Omega^1_{E/Y}$ with a locally free resolution locally isomorphic to $0\to\mathcal{O}_\widetilde{X}^d(-E)\to\mathcal{O}_\widetilde{X}^d\to 0$, we see the identification $\mathcal{Ext}^1(i_*\Omega^1_{E/Y},\mathcal{O}_{\widetilde{X}})\cong i_*T_{E/Y}(E)$. So our long exact sequence reduces to $$0\to T_{\widetilde{X}}\to \pi^*T_X\to i_*T_{E/Y}(E)\to 0.$$
This is the correct relationship between $T_X$ and $T_\widetilde{X}$. It's completely independent of $Y$ as long as $Y$ is a smooth subvariety.
The correct setting for an equation like you ask for in (2) to hold is with the canonical sheaf of a blowup. Per Griffiths and Harris, for example, $\omega_{\widetilde{X}} \cong \pi^*\omega_X\otimes \mathcal{O}((c-1)E)$ where $c$ is the codimension of $Y$ in $X$ and $\omega_X=\bigwedge^{\dim{X}} \Omega^1_X$. Translating this in to tangent sheaves by dualizing and using the fact that dualizing commutes with taking determinant bundles (plus that $(\mathcal{O}(mE))^*=\mathcal{O}(-mE)$), we see that we should have $\bigwedge^nT_\widetilde{X}\cong (\bigwedge^nT_X)((1-c)E)$. There is no relation between $c$ and $\deg Y$.
