# Why is this bundle positive?

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 $$X$$.

• In what context do you see this problem? Jul 28, 2021 at 16:37
• It comes from Claire Voisin's book Hodge theory and complex algebraic geometry; chapter 7 exercice 2)d) page 173. Jul 28, 2021 at 16:43

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.