Given an ample line bundle on a curve is there any chance for its pullback to the projective bundle of some vector bundle to be big? It is known that first cohomology of inverse of nef and big line bundles can be killed by Frobenius if the dimension of variety $\geq 2$. I was wondering whether such a vanishing can hold for pullbacks of ample line bundles to projective bundles.
Is it possible for the pullback of an ample line bundle under projection to be big
algebraic-geometryline-bundlessheaf-cohomologyvector-bundles
Related Solutions
Let $X$ be your projective variety, and let $\mathcal O(1)$ be a very ample sheaf on $X$ associated to a closed immersion of $X$ into $\mathbb P_k^r$.
Because $\mathcal O(1)$ is ample (since it is very ample) and invertible sheaves are coherent, for every $i$ you have that there exists an integer $m_i > 0$ such that for all $m > m_i$, the sheaf $\mathcal L_i \otimes \mathcal O(m)$ is generated by global sections (here, $\mathcal L_i$ is the invertible sheaf corresponding to the line bundle $L_i$)
Let now $M$ be some integer larger than all the $m_i$. Then for all $i$, the sheaves $\mathcal L_i \otimes \mathcal O(M)$ are generated by global sections.
By exercise II.7.5d) in Hartshorne's Algebraic Geometry (see for example this post), the sheaves $\mathcal L_i \otimes \mathcal O(M + 1)$ are very ample.
Thus, if you take $B$ to be the very ample line bundle associated to the very ample sheaf $\mathcal O(M + 1)$, you have that $B \otimes L_i$ is very ample for all $i$.
The right reference for this material is probably Lazarsfeld's book Positivity in Algebraic Geometry, Volume $1$, and this is a combination of results. (I'd argue it does not follow immediately from Riemann-Roch.) This book is also written in the language of algebraic geometry, but it at least contains this result. See the end of section 1.4 for example.
For a vector bundle $E$ we can define the Euler characteristic $\chi(M, E) = \sum_i (-1)^i h^i(M, E)$. Riemann-Roch theorems seek to compute this using the intersection theory of $M$ and $E$. For our purpose, we only need the following version of the asymptotic Riemann-Roch theorem.
Theorem: Let $L$ be a line bundle on (a smooth projective variety) $M$. Then, $$\chi(M, L^m) = Cm^n + O(m^{n - 1})$$ is a polynomial in $m$ of degree $\leq n$, where $C = \int_M c_1(L)^n$.
When $L$ is nef we also have the following fact which controls the asymptotic growth of the higher cohomology.
Theorem: Let $L$ be a nef line bundle on $M$. Then $$h^i(M, L^m) = O(m^{n - i})$$ for $m$ sufficiently large and $i \geq 0$.
Together, these theorems imply that for a nef line bundle, we can describe the asymptotic growth $h^0(M, L^m)$ to be $$h^0(M, L^m) = \left (\int_M c_1(L)^n\right)m^n + O(m^{n - 1})$$ for $m$ sufficiently large. In particular, $L$ is big if and only if this first coefficient is positive.
Both these theorems are nontrivial and require a significant amount of algebraic geometry to prove. The proof of the second fact is given under theorem 1.4.40 in Lazarsfeld's book, and uses vanishing theorems. The asymptotic Riemann-Roch theorem is stated in Lazarsfeld's book but a proof can only be found elsewhere. (Olivier Debarre's Higher Dimensional Algebraic Geometry, for example.) Both these books assume and use a lot of the language algebraic geometry, though.
Best Answer
No. Just use the projection formula to compute the cohomology of powers of the pullback.