What is known about the cohomology of the relative tangent bundle on a conic bundle?

Question

Let us assume $X$ is a smooth, projective and unirational variety of dimension $n$ over $\mathbb{C}$.

Given a conic bundle $\pi: Y\rightarrow X$ such that $\omega_{\pi}^{-1}$ is relatively very ample with respect to $\pi$, here $\omega_{\pi}=\omega_Y\otimes \pi^{*}\omega_X^{-1}$. Assume all three possible types of fibers occur, that is we have either smooth conics $(\cong \mathbb{P}^1)$, pairs of intersecting lines $(\cong \mathbb{P}^1$ v $\mathbb{P}^1)$ or nonreduced double lines $(\cong 2\mathbb{P}^1)$

Is anything known about the cohomology of the relative tangent bundle of $\pi$, i.e. what is $H^i(Y,\mathcal{T}_{\pi})$ for $0\leq i\leq n+1$, where $\mathcal{T}_{\pi}=\omega_{\pi}^{-1}$?

I am interested in any information about these groups. I am especially interested in the groups for $i=0,1,2$. If it helps, one can assume that $\pi$ is a standard conic bundle, that is $Y$ is also smooth and we have $Pic(Y)=\mathbb{Z}K_Y\oplus\pi^{*}Pic(X)$ and $\pi^{-1}(C)$ is irreducible for any irreducible curve $C\subset X$. Here $K_Y$ is the canonical divisor of $Y$.

Answer 1

Such conic bundle is given by a rank three vector bundle, say $E$ on $X$, and a line subbundle $L \subset Sym^2E$ (just take $E$ to be the pushforward of $\omega^{-1}_\pi$, and $L$ corresponds to the equation of the conic bundle). Then the pushforward of $T_\pi$ to $X$ is isomorphic to $L \otimes E \otimes \det E^\vee$ (and derived pushforwards vanish). A priori this vector bundle on $X$ may have more or less any cohomology (except, of course, for the vanishing of the cohomology in degree $n+1$).