# Iitaka dimensions of vector bundles

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## Abstract

Let 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 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 An immediate corollary of this inequality is that if

*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$$\begin{aligned} \phi _m:X\dashrightarrow Y_m\subseteq {\mathbb {P}}\bigl (H^0(X,L^{\otimes m})\bigr ), \end{aligned}$$

*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$$\begin{aligned} \varphi _m:X\dashrightarrow Y_m\subseteq {\mathbb {G}}(H^0(X,S^m E),{{\mathrm{rank}}}S^m E), \end{aligned}$$

*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$$\begin{aligned} \kappa (E)\ge \kappa \bigl ({\mathcal {O}}_{{\mathbb {P}}(E)}(1)\bigr )-{{\mathrm{rank}}}E+1. \end{aligned}$$

*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.Lazarsfeld, R.: Positivity in Algebraic Geometry I–II, Ergeb. Math. Grenzgeb, vols. 48–49. Springer, Berlin (2004)Google Scholar
- 2.Mistretta, E., Urbinati, S.: Iitaka fibrations for vector bundles, IMRN, rnx239. https://doi.org/10.1093/imrn/rnx239

## Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2018