Annali di Matematica Pura ed Applicata (1923 -)

, Volume 197, Issue 5, pp 1631–1635

# Iitaka dimensions of vector bundles

• Shin-Yao Jow
Article

## Abstract

Let X be a projective variety. If L is a line bundle on X, for each positive integer m in $${\mathbf {N}}(L)=\{m\in {\mathbb {N}}\mid H^0(X,L^{\otimes m})\ne 0\}$$, the global sections of $$L^{\otimes m}$$ define a rational map
\begin{aligned} \phi _m:X\dashrightarrow Y_m\subseteq {\mathbb {P}}\bigl (H^0(X,L^{\otimes m})\bigr ), \end{aligned}
where $$Y_m$$ is the closure of $$\phi _m(X)$$. It is well-known that for all sufficiently large $$m\in {\mathbf {N}}(L)$$, the rational maps $$\phi _m:X\dashrightarrow Y_m$$ are birationally equivalent to a fixed fibration (the Iitaka fibration), and $$\kappa (L):=\dim Y_m$$ is called the Iitaka dimension of L. In a recent paper titled “Iitaka fibrations for vector bundles”, Mistretta and Urbinati generalized this to a vector bundle E on X. Let $${\mathbf {N}}(E)$$ be the set of positive integers m such that the evaluation map $$H^0(X,S^m E)\rightarrow S^m E_x$$ is surjective for all points x in some nonempty open subset of X. For each $$m\in {\mathbf {N}}(E)$$, the global sections of $$S^m E$$ define a rational map
\begin{aligned} \varphi _m:X\dashrightarrow Y_m\subseteq {\mathbb {G}}(H^0(X,S^m E),{{\mathrm{rank}}}S^m E), \end{aligned}
where $${\mathbb {G}}(H^0(X,S^m E),{{\mathrm{rank}}}S^m E)$$ is the Grassmannian of $${{\mathrm{rank}}}S^m E$$-dimensional quotients of $$H^0(X,S^m E)$$. Mistretta and Urbinati showed that for every $$m\in {\mathbf {N}}(E)$$, the rational maps $$\varphi _{km}$$ are birationally equivalent for sufficiently large k, and called $$\kappa (E):=\dim Y_{km}$$ the Iitaka dimension of E. Here we first slightly improve Mistretta and Urbinati’s result to show that the rational maps $$\varphi _{m}$$ are birationally equivalent for all sufficiently large $$m\in {\mathbf {N}}(E)$$. Then we show that
\begin{aligned} \kappa (E)\ge \kappa \bigl ({\mathcal {O}}_{{\mathbb {P}}(E)}(1)\bigr )-{{\mathrm{rank}}}E+1. \end{aligned}
An immediate corollary of this inequality is that if E is big then $$\kappa (E)=\dim X$$, which answers a question of Mistretta and Urbinati. Another corollary is that if E is big then $$\det E$$ is big, provided that $${\mathbf {N}}(E)\ne \emptyset$$.

## Keywords

Asymptotically generically generated Big vector bundle Iitaka dimension

## Mathematics Subject Classification

14C20 14J60 32L10

## Notes

### Acknowledgements

The author gratefully acknowledges the support of MoST (Ministry of Science and Technology, Taiwan).

## References

1. 1.
Lazarsfeld, R.: Positivity in Algebraic Geometry I–II, Ergeb. Math. Grenzgeb, vols. 48–49. Springer, Berlin (2004)Google Scholar
2. 2.
Mistretta, E., Urbinati, S.: Iitaka fibrations for vector bundles, IMRN, rnx239.