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
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
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
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 \({\mathbf {N}}(E)\ne \emptyset \).
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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)
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)
E is generically generated if it is globally generated on some nonempty open subset of X.
-
(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
Then one can define a rational map
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
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
That is, there exists a commutative diagram
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
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
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
are surjective for each \(x\in U\). It thus follows from the commutative diagram
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
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
Pick any \(w\in H^0(X,S^q E)\) such that
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
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
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
so there are projection morphisms
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
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
be the morphism defined by the global sections of \({\mathcal {O}}_{{\mathbb {P}}(E)}(m)\) on \({\mathbb {P}}(E)\), and let
Then there is a commutative diagram
For each point \(x\in U\),
and
is the mth Veronese embedding. Hence
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 \(m\in {\mathbf {N}}(E)\),
\(\square \)
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
If E is big (and AGG), \(\kappa (E)=\dim X\) by Corollary 5, and hence \(\kappa (\det E)=\dim X\).
References
Lazarsfeld, R.: Positivity in Algebraic Geometry I–II, Ergeb. Math. Grenzgeb, vols. 48–49. Springer, Berlin (2004)
Mistretta, E., Urbinati, S.: Iitaka fibrations for vector bundles, IMRN, rnx239. https://doi.org/10.1093/imrn/rnx239
Acknowledgements
The author gratefully acknowledges the support of MoST (Ministry of Science and Technology, Taiwan).
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Jow, SY. Iitaka dimensions of vector bundles. Annali di Matematica 197, 1631–1635 (2018). https://doi.org/10.1007/s10231-018-0740-1
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DOI: https://doi.org/10.1007/s10231-018-0740-1