Let $E$ be a holomorphic bundle (with rank $r$) over a compact complex manifold $X$. $E^{*}$ will denote its dual. $\mathbb{P}(E^{*})$ will denote the projective bundle associated to the dual of $E$ and $O(1)$ will be the dual bundle of the tautological bundle over $\mathbb{P}(E^{*})$ (cf. here: https://en.wikipedia.org/wiki/Projective_bundle#CITEREFHartshorne ).
I consider $\pi: \mathbb{P}(E^{*}) \mapsto X$ the projection and I would like to show
$$\pi^{*}K_{X}^{*} \otimes \pi^{*}\det(E^{*}) \otimes O(1)^{r+1} \otimes \pi^{*}L^{\otimes N}$$
is positive over $\mathbb{P}(E^{*})$ for $N$ large enough if $L$ is a positive line bundle (i.e. if $L$ admits a hermitian form whose curvature form is a Kahler form on X).
Do you have any idea to prove it?
PS : $\det(E)$ is the line bundle of the holomorphic $r$ forms over $E$ and $K_X$ the
bundle of the holomorphic $\dim(X)$ form over $X$.
Best Answer
In general it's a fact that on any projective variety $M\otimes L^{\otimes N}$ is a positive line bundle for line bundles $M$ and $L$ with $L$ positive and $N \gg 0$ (because the curvature forms add when tensoring line bundles).
Next, it's another general fact that if $\pi : Y \rightarrow X$ is a projective morphism, $M$ a positive line bundle on $Y$, and $L$ a positive line bundle on $X$, then $M\otimes \pi^*L$ is positive (because the positive curvature form will pull back to at least a non-negative one).
Now, the bundle in question can be rewritten as: $$ (\mathcal{O}(1)^{r+1}\otimes \pi^*L^{N_1})\otimes \pi^*(K_X^*\otimes \text{det}(E^*)\otimes L^{N_2}) $$ for $N_1,N_2 \gg 0$ and $N = N_1 + N_2$. The bundle $K_X^*\otimes \text{det}(E^*)\otimes L^{N_2}$ is then positive by the first fact (since $L$ is positive), so by the second fact we are reduced to showing that $\mathcal{O}(1)^{r+1}\otimes \pi^*L^{N_1}$ is positive.
To see (somewhat less precisely) that $\mathcal{O}(1)^{r+1}\otimes \pi^*L^{N_1}$ is positive, note that $\mathcal{O}(1)$ restricts to $\mathcal{O}_{\mathbb{P}^r}(1)$ on the fibers of $\pi$ so it is positive in directions tangent to those fibers. And $\pi^*L^{N_1}$ is as positive as you like in all other directions. So $\mathcal{O}(1)^{r+1}\otimes \pi^*L^{N_1}$ is positive in all directions.