Iitaka dimensions of vector bundles

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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.  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

Authors and Affiliations

  1. 1.Department of MathematicsNational Tsing Hua UniversityHsinchu CityTaiwan

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