1 Introduction

Notations and conventions We will work over an algebraically closed field. Varieties are assumed to be irreducible. Points of a variety mean closed points. If V is a vector space, \({\mathbb {P}}(V)\) denotes the projective space of one-dimensional quotients of V, and \({\mathbb {G}}(V,r)\) denotes the Grassmannian of r-dimensional quotients of V. If E is a vector bundle on a variety X and \(x\in X\) is a point, we denote by \({\mathbb {P}}(E)\) the projective bundle of one-dimensional quotients of E, and by \(E_x\) the fiber of E over x (not the stalk of the germs of sections of E at x).

Let X be a projective variety. It is a basic construction in algebraic geometry that a line bundle L on X such that \(H^0(X,L^{\otimes m})\ne 0\) for some \(m>0\) naturally induces a rational map from X to the projective space \({\mathbb {P}}\bigl (H^0(X,L^{\otimes m})\bigr )\). In [2], Mistretta and Urbinati studied a generalization of this construction to vector bundles as follows.

Definition 1

Let E be a vector bundle on a projective variety X, and let U be an open subset of X. We say that

  1. (1)

    E is globally generated on U if the evaluation map \(H^0(X,E)\rightarrow E_x\) is surjective for every point \(x\in U\).

  2. (2)

    E is generically generated if it is globally generated on some nonempty open subset of X.

  3. (3)

    E is asymptotically generically generated (AGG) if for some positive integer m, the mth symmetric power \(S^m E\) of E is generically generated.

Definition 2

Let E be an AGG vector bundle of rank r on a projective variety X. Let m be a positive integer such that \(S^m E\) is globally generated on a nonempty open subset \(U\subseteq X\). Denote

$$\begin{aligned} \sigma _m(r)={{\mathrm{rank}}}S^m E=\left( {\begin{array}{c}m+r-1\\ m\end{array}}\right) . \end{aligned}$$

Then one can define a rational map

$$\begin{aligned} \varphi _m:X\dashrightarrow {\mathbb {G}}\bigl (H^0(X,S^m E),\sigma _m(r)\bigr ) \end{aligned}$$

by sending a point \(x\in U\) to the \(\sigma _m(r)\)-dimensional quotient \([H^0(X,S^m E)\twoheadrightarrow S^m E_x]\) of \(H^0(X,S^m E)\) under the evaluation map. We call \(\varphi _m\) the mth Kodaira map of E.

Our first result is a slight improvement on [2, Theorem 4.4].

Theorem 3

Let E be an AGG vector bundle on a complex projective variety X, and denote

$$\begin{aligned} {\mathbf {N}}(E)=\{m\in {\mathbb {N}}\mid S^mE\text { is generically generated}\}. \end{aligned}$$

For each \(m\in {\mathbf {N}}(E)\), let \(\varphi _m\) be the mth Kodaira map of E, and let \(Y_m\) be the closure of \(\varphi _m(X)\). Then for all sufficiently large \(m\in {\mathbf {N}}(E)\), the rational maps \(\varphi _{m}:X\dashrightarrow Y_{m}\) are birationally equivalent to a fixed surjective morphism of projective varieties

$$\begin{aligned} \varphi _{\mathbb {G}}:X_{\mathbb {G}}\rightarrow Y_{\mathbb {G}}. \end{aligned}$$

That is, there exists a commutative diagram

figure a

where the horizontal maps are birational and \(u_{\mathbb {G}}\) is a morphism.

The statement in [2, Theorem 4.4] is the same except that instead of “for all sufficiently large \(m\in {\mathbf {N}}(E)\)”, they have “for every \(m\in {\mathbf {N}}(E)\) and for \(k\gg 0\)” the rational maps \(\varphi _{km}:X\dashrightarrow Y_{km}\) are birationally equivalent to a fixed surjective morphism \(\varphi _{\mathbb {G}}:X_{\mathbb {G}}\rightarrow Y_{\mathbb {G}}\). Our version is more in line with the original Iitaka fibrations for line bundles [1, Theorem 2.1.33]. We also remark that compared to the case of line bundles, the difficulty in the proof comes from the fact that for a vector bundle E, \(S^p E\otimes S^q E\) is not isomorphic to \(S^{p+q}E\), and also that the morphism \(\varphi _{\mathbb {G}}:X_{\mathbb {G}}\rightarrow Y_{\mathbb {G}}\) is in general not a fibration (i.e., may not have connected fibers: see [2, Example 3.7]).

Following [2], we call the dimension of \(Y_{\mathbb {G}}\) the Iitaka dimension of E, and denote it by \(\kappa (X,E)\) or \(\kappa (E)\). Mistretta and Urbinati showed that if E is strongly semiample, meaning that \(S^m E\) is globally generated on X for some \(m>0\), then \(\kappa (E)=\kappa (\det E)\) [2, Remark 4.5]. They then raised several questions on Iitaka dimensions for vector bundles which are not necessarily strongly semiample. Our second result, which is an inequality relating the Iitaka dimensions of E and \({\mathcal {O}}_{{\mathbb {P}}(E)}(1)\), can help answer some of those questions.

Theorem 4

Let E be an AGG vector bundle of rank r on a projective variety X. Let \(\pi :{\mathbb {P}}(E)\rightarrow X\) be the projective bundle of one-dimensional quotients of E, and let \({\mathcal {O}}_{{\mathbb {P}}(E)}(1)\) be the tautological quotient line bundle of \(\pi ^*E\) on \({\mathbb {P}}(E)\). Then

$$\begin{aligned} \kappa (X,E)\ge \kappa \bigl ({\mathbb {P}}(E),{\mathcal {O}}_{{\mathbb {P}}(E)}(1)\bigr )-r+1. \end{aligned}$$

Corollary 5

Let E be an AGG vector bundle on a projective variety X. If E is big (meaning that \({\mathcal {O}}_{{\mathbb {P}}(E)}(1)\) is big), then \(\kappa (E)=\dim X\).

This answers Question 4.7 and 4.8 in [2].

Corollary 6

Let E be an AGG vector bundle on a projective variety X. If E is big, then \(\det E\) is big.

This answers Question 4.6 in [2] affirmatively when E is big. Note that Corollary 6 is interesting in its own right since it is false without the AGG assumption: for example let E be a direct sum of line bundles

$$\begin{aligned} E={\mathcal {O}}_X(D_1)\oplus \cdots \oplus {\mathcal {O}}_X(D_r). \end{aligned}$$

Then E is big if and only if some nonnegative \({\mathbb {Z}}\)-linear combination of the \(D_i\) is big [1, Lemma 2.3.2], whereas \(\det E\) is big if and only if \(D_1+\cdots +D_r\) is big.

2 Proofs

Lemma 7

Let E be a vector bundle on a projective variety X. If p and q are positive integers such that \(S^p E\) and \(S^q E\) are globally generated on an open subset U of X, then \(S^{p+q} E\) is globally generated on U. Moreover, if \(x,y\in U\) are points such that \(\varphi _p(x)\ne \varphi _p(y)\), then \(\varphi _{p+q}(x)\ne \varphi _{p+q}(y)\).

Proof

By assumption the evaluation maps

$$\begin{aligned} {{\mathrm{ev}}}_{p,x}:H^0(X,S^p E)\twoheadrightarrow S^p E_x\ \text { and }\ {{\mathrm{ev}}}_{q,x}:H^0(X,S^q E)\twoheadrightarrow S^q E_x \end{aligned}$$

are surjective for each \(x\in U\). It thus follows from the commutative diagram

figure b

that the evaluation map \(H^0(X,S^{p+q} E)\rightarrow S^{p+q} E_x\) is surjective. Hence \(S^{p+q} E\) is globally generated on U.

For each point \(x\in U\), by definition

$$\begin{aligned} \varphi _p(x)=[{{\mathrm{ev}}}_{p,x}:H^0(X,S^p E)\twoheadrightarrow S^p E_x]. \end{aligned}$$

If \(x,y\in U\) are points such that \(\varphi _p(x)\ne \varphi _p(y)\), then there exists an element v in \(H^0(X,S^p E)\) such that

$$\begin{aligned} {{\mathrm{ev}}}_{p,x}(v)=0\ \text { and }\ {{\mathrm{ev}}}_{p,y}(v)\ne 0. \end{aligned}$$

Pick any \(w\in H^0(X,S^q E)\) such that

$$\begin{aligned} {{\mathrm{ev}}}_{q,x}(w)\ne 0\ \text { and }\ {{\mathrm{ev}}}_{q,y}(w)\ne 0. \end{aligned}$$

Denote by \(v\cdot w\in H^0(X,S^{p+q} E)\) the image of \(v\otimes w\in H^0(X,S^p E)\otimes H^0(X,S^q E)\) under the multiplication map. It follows from the commutative diagram above that

$$\begin{aligned} {{\mathrm{ev}}}_{p+q,x}(v\cdot w)= 0\ \text { and }\ {{\mathrm{ev}}}_{p+q,y}(v\cdot w)\ne 0. \end{aligned}$$

Hence \(\varphi _{p+q}(x)\ne \varphi _{p+q}(y)\). \(\square \)

Proof of Theorem 3

By the proof of [2, Theorem 4.4], for each \(m\in {\mathbf {N}}(E)\) the rational map \(\varphi _m:X\dashrightarrow Y_m\) factors through \(\varphi '_{\mathbb {G}}=\varphi _{\mathbb {G}}\circ u_{\mathbb {G}}^{-1}:X\dashrightarrow Y_{\mathbb {G}}\). So to show that \(\varphi _m=\varphi '_{\mathbb {G}}\) for sufficiently large m, it is enough to show that \(\varphi '_{\mathbb {G}}(x)\ne \varphi '_{\mathbb {G}}(y)\) implies \(\varphi _m(x)\ne \varphi _m(y)\) for all points x and y in some nonempty open subset of X.

Pick \(m_1,\ldots ,m_n\in {\mathbf {N}}(E)\) which generate \({\mathbf {N}}(E)\) as a semigroup. By [2, Theorem 4.4], \(\varphi _{km_i}=\varphi '_{\mathbb {G}}\) for \(k\gg 0\). Hence all sufficiently large \(m\in {\mathbf {N}}(E)\) can be written as \(m=\sum _{i=1}^{n} k_im_i\), where \(k_i\in {\mathbb {N}}\) and at least one of \(\varphi _{k_im_i}=\varphi '_{\mathbb {G}}\). It thus follows from Lemma 7 that \(\varphi '_{\mathbb {G}}(x)\ne \varphi '_{\mathbb {G}}(y)\) implies \(\varphi _m(x)\ne \varphi _m(y)\). \(\square \)

Proof of Theorem 4

Let \(m>0\) be a positive integer such that \(S^m E\) is globally generated on a nonempty open subset U of X. We denote

$$\begin{aligned} V=H^0(X,S^m E)\ \text { and }\ {\mathbb {G}}_m={\mathbb {G}}\bigl (V,\sigma _m(r)\bigr ). \end{aligned}$$

Let \(Q_m\) be the tautological quotient bundle on \({\mathbb {G}}_m\). Then \({\mathbb {P}}(Q_m)\) is isomorphic to a two-step flag variety

$$\begin{aligned} {\mathbb {P}}(Q_m)=\{[V\twoheadrightarrow W_1\twoheadrightarrow W_2]\mid \dim W_1=\sigma _m(r),\dim W_2=1\}, \end{aligned}$$

so there are projection morphisms

$$\begin{aligned} \pi _1:{\mathbb {P}}(Q_m)\rightarrow {\mathbb {G}}_m \ \text { and }\ \pi _2:{\mathbb {P}}(Q_m)\rightarrow {\mathbb {P}}(V). \end{aligned}$$

Let \(x\in U\) be a point. Then the evaluation map \(H^0(X,S^m E)\twoheadrightarrow S^m E_x\) is surjective. By definition a point \(y\in \pi ^{-1}(x)\) is a one-dimensional quotient \([E_x\twoheadrightarrow L]\) of \(E_x\). Since

$$\begin{aligned} H^0\bigl ({\mathbb {P}}(E),{\mathcal {O}}_{{\mathbb {P}}(E)}(m)\bigr )=V=H^0(X,S^m E)\twoheadrightarrow S^m E_x\twoheadrightarrow S^m L={\mathcal {O}}_{{\mathbb {P}}(E)}(m)_y, \end{aligned}$$

the line bundle \({\mathcal {O}}_{{\mathbb {P}}(E)}(m)\) is globally generated on the open set \({\widetilde{U}}=\pi ^{-1}(U)\subseteq {\mathbb {P}}(E)\). Let

$$\begin{aligned} \phi _m:{\widetilde{U}}\rightarrow {\mathbb {P}}(V),\quad y=[E_x\twoheadrightarrow L]\mapsto [V\twoheadrightarrow S^m L] \end{aligned}$$

be the morphism defined by the global sections of \({\mathcal {O}}_{{\mathbb {P}}(E)}(m)\) on \({\mathbb {P}}(E)\), and let

$$\begin{aligned} \psi _m:{\widetilde{U}}\rightarrow {\mathbb {P}}(Q_m),\quad y=[E_x\twoheadrightarrow L]\mapsto [V\twoheadrightarrow S^m E_x\twoheadrightarrow S^m L]. \end{aligned}$$

Then there is a commutative diagram

figure c

For each point \(x\in U\),

$$\begin{aligned} \pi ^{-1}(x)={\mathbb {P}}(E_x),\ \pi _1^{-1}\bigl (\varphi _m(x)\bigr )={\mathbb {P}}(S^m E_x), \end{aligned}$$

and

$$\begin{aligned} \psi _m\bigr |_{\pi ^{-1}(x)}:{\mathbb {P}}(E_x)\rightarrow {\mathbb {P}}(S^m E_x),\quad [E_x\twoheadrightarrow L]\mapsto [S^m E_x\twoheadrightarrow S^m L] \end{aligned}$$

is the mth Veronese embedding. Hence

$$\begin{aligned} \dim \varphi _m(U)=\dim \psi _m({\widetilde{U}})-r+1\ge \dim \phi _m({\widetilde{U}})-r+1. \end{aligned}$$

Since \(\dim \varphi _m(U)=\kappa (X,E)\) and \(\dim \phi _m({\widetilde{U}})=\kappa \bigl ({\mathbb {P}}(E),{\mathcal {O}}_{{\mathbb {P}}(E)}(1)\bigr )\) for sufficiently large \(m\in {\mathbf {N}}(E)\),

$$\begin{aligned} \kappa (X,E)\ge \kappa \bigl ({\mathbb {P}}(E),{\mathcal {O}}_{{\mathbb {P}}(E)}(1)\bigr )-r+1. \end{aligned}$$

\(\square \)

Corollary 5 follows immediately from Theorem 4. As for Corollary 6, applying the argument in the proof of [2, Theorem 3.4] to an open set \(U\subseteq X\) where \(S^m E\) is globally generated, one sees that the rational map \(\varphi _{\mathbb {G}}:X\dashrightarrow Y_{\mathbb {G}}\) is a composition of the Iitaka fibration \(\varphi _{\det E}:X\dashrightarrow Y_{\infty }\) followed by a dominant rational map \(Y_{\infty }\dashrightarrow Y_{\mathbb {G}}\). It follows that

$$\begin{aligned} \kappa (\det E)\ge \kappa (E). \end{aligned}$$

If E is big (and AGG), \(\kappa (E)=\dim X\) by Corollary 5, and hence \(\kappa (\det E)=\dim X\).