A weak criterion of bigness for toric vector bundles

TL;DR

This paper introduces a relaxed criterion for determining when equivariant vector bundles on toric varieties are big, advancing the understanding of their geometric properties.

Contribution

It presents a weaker version of the bigness criterion specifically for equivariant vector bundles on toric varieties, expanding theoretical tools.

Findings

Established a weak bigness criterion for equivariant vector bundles

Extended the theoretical framework for vector bundles on toric varieties

Provided conditions under which vector bundles are considered big

Abstract

We prove a weak version of a bigness criterion for equivariant vector bundles on toric varieties.

Equations

$${\mathcal{F}}_{\rho }(\mathcal{E})=\{E\supseteq \cdots \supseteq {\mathcal{E}}^{\rho }(j)\supseteq {\mathcal{E}}^{\rho }(j+1)\supseteq \cdots \supseteq (0)\},\rho \in P,$$

$${\mathcal{E}}^{\rho }(j)=\underset{\begin{array}{c}\chi \in M(T_{\sigma })\\ (\chi ,v_{\rho })\ge j\end{array}}{⨁}{E}_{\chi }^{\sigma }.$$

$${\mathcal{E}}^n(j)=\underset{\begin{array}{c}\chi \in M(T_{\sigma })\\ (\chi ,n)\ge j\end{array}}{⨁}{E}_{\chi }^{\sigma }\text{if}n\in |\sigma |.$$

$${\varphi }_e(n)=\underset{j}{\max }\{j\mid e\in {\mathcal{E}}^n(j)\},n\in N,\text{and}$$

$${\Delta }_e=\{u\in M{\otimes }_{\mathbb{Z}}\mathbb{Q}\mid (u,v_{\rho })\le {\varphi }_e(v_{\rho }),\rho \in P\}.$$

$$H^0(X,\mathcal{E})=\underset{u\in M}{⨁}\mathrm{Span}\{{\chi }^{-u}\otimes e\mid u\in {\Delta }_e\}\subset k[M]{\otimes }_{k}E\text{cite}[cite][\text{@@bibref}Klyachko].$$

$$H^0(X,\mathcal{E})=\underset{e\in ϵ(X,\mathcal{E})}{\mathrm{Span}}\{{\chi }^{-u}\otimes e\mid u\in {\Delta }_e\cap M\}.$$

$$\underset{l\to \infty }{\mathrm{limsup}}\frac{\dim \mathrm{Im}(S^l{H}^0(X,{\mathrm{Sym}}^p\mathcal{E})\to H^0(X,{\mathrm{Sym}}^{pl}\mathcal{E}))}{(pl{)}^d/d!}>0,$$

$$\underset{l\to \infty }{\mathrm{limsup}}\left(\frac{1}{l^d}\cdot \dim \underset{e_1,\dots ,e_l\in ϵ(X,{\mathrm{Sym}}^p\mathcal{E})}{\mathrm{Span}}\{{\chi }^{-(u_1+\dots +u_l)}\otimes (e_1\cdots e_l)\mid u_i\in {\Delta }_{e_i}\cap M,i=1,\dots ,l\}\right)>0.$$

$${\left({\mathrm{Sym}}^p\mathcal{E}\right)}^{\rho }(j)=\sum _{j_1+\dots +j_p=j}{\mathcal{E}}^{\rho }(j_1)\cdot \dots \cdot {\mathcal{E}}^{\rho }(j_p)\subset {\mathrm{Sym}}^{p}E,j\in \mathbb{Z},\rho \in P.$$

$$c_1\cdot {\Delta }_{f_1}+\dots +c_q\cdot {\Delta }_{f_q}\subset {\Delta }_f,$$

$$L(X,\mathcal{E})=\mathrm{Span}\{a-b\mid a,b\in {\Delta }_f,f\in {\mathrm{Sym}}^{p}E\setminus \{0\},p>0\}\subset M{\otimes }_{\mathbb{Z}}\mathbb{Q}.$$

$$\Lambda =\{w({\Delta }_f)\mid f\in {\mathrm{Sym}}^{p}E\setminus \{0\},p>0,{\Delta }_f\ne \varnothing \}\subset M{\otimes }_{\mathbb{Z}}\mathbb{Q}/L(X,\mathcal{E}).$$

$$w_f=w({\Delta }_f),f\in \overline{ϵ}(X,{\mathrm{Sym}}^p\mathcal{E}).$$

$$\mathcal{A}(p,X,\mathcal{E})=\underset{\overline{w}\in {\Lambda }_p}{⨁}{\mathcal{A}}_{\overline{w}}(p,X,\mathcal{E}),$$

$${\mathcal{A}}_{(w,l)}(p,X,\mathcal{E})=\mathrm{Span}\{f_1^{c_1}\dots f_q^{c_q}\mid c_i\ge 0,c_1\cdot w_{f_1}+\dots +c_q\cdot w_{f_q}=w,c_1+\dots +c_q=l\}\subset {\mathrm{Sym}}^{pl}E.$$

$$\alpha (p,X,\mathcal{E})=\underset{l\to \infty }{\mathrm{limsup}}\sum _{w\in \Lambda ,(w,l)\in {\Lambda }_p}\dim {\mathcal{A}}_{(w,l)}(p,X,\mathcal{E})/l^{\dim X-\dim L(X,\mathcal{E})+\mathrm{rk}\mathcal{E}-1}\in \mathbb{R}.$$

$$\bigcup _{p>0}ϵ(X,{\mathrm{Sym}}^p\mathcal{E})\subset \underset{p\ge 0}{⨁}{\mathrm{Sym}}^{p}E=\mathrm{Sym}E$$

$$0<\underset{l\to \infty }{\mathrm{limsup}}\left(\frac{1}{l^d}\cdot \dim \underset{\begin{array}{c}{c}_1+\dots +c_q=l\\ c_i\ge 0\end{array}}{\mathrm{Span}}\{{\chi }^{-u}\otimes f_1^{c_1}\cdots f_q^{c_q}\mid u\in \left(c_1\cdot {\Delta }_{f_1}+\dots +c_q\cdot {\Delta }_{f_q}\right)\cap M\}\right).$$

$$\left(\underset{l\to \infty }{\mathrm{limsup}}\frac{\#\left(M\cap l\cdot \left({\Delta }_{f_1}+\dots +{\Delta }_{f_q}\right)\right)}{l^{\dim L(X,\mathcal{E})}}\right)\cdot \alpha (p,X,\mathcal{E}),$$

$$\bigcap _{\begin{array}{c}{c}_1,\dots ,c_q\in \mathbb{R}\\ {\max }_i|c_i|\le 1/N\end{array}}(1+c_1)\cdot {\tilde{\Delta }}_{f_1}+\dots +(1+c_q)\cdot {\tilde{\Delta }}_{f_q}$$

$$\#\left(M\cap \bigcap _{\begin{array}{c}f=(f_1\dots f_q{)}^{lN}\cdot f_1^{c_1}\dots f_q^{c_q}\\ c_i\ge 0,{\sum }_i{c}_i\cdot w_{f_i}=w,{\sum }_i{c}_i=l\end{array}}{\Delta }_f\right)\ge C\cdot l^{\dim L(X,\mathcal{E})},$$

$$\underset{l\to \infty }{\mathrm{limsup}}\frac{\dim H^0(X,{\mathrm{Sym}}^{p(Nql+l)}\mathcal{E})}{(Nql+l{)}^d}\ge$$

$$\ge \underset{l\to \infty }{\mathrm{limsup}}\left(\frac{1}{((Nq+1)l{)}^d}\cdot \dim \underset{\begin{array}{c}{c}_1+\dots +c_q=l\\ c_i\ge 0\end{array}}{\mathrm{Span}}\{{\chi }^{-u}\otimes f\mid u\in {\Delta }_f\cap M,f=(f_1\dots f_q{)}^{lN}\cdot f_1^{c_1}\dots f_q^{c_q}\}\right)\ge$$

$$\ge \frac{C}{(Nq+1{)}^d}\cdot \alpha (p,X,\mathcal{E})>0,$$

$$\bigcup _{p>0}ϵ(X,{\mathrm{Sym}}^p\mathcal{E})\subset \mathrm{Sym}E$$

$$\underset{w\in \Lambda ,(w,l)\in {\Lambda }_p}{⨁}{\mathcal{A}}_{(w,l)}(p,X,\mathcal{E})\subset {\mathrm{Sym}}^{pl}E,l>0.$$

Full text

1 Preliminaries

Let $k$ be an algebraically closed field of characteristic [math], $T=(k^{*})^{r}$ a torus, $M=M(T)\cong{\mathbb{Z}}^{r}$ and $N={M(T)}^{\vee}$ the character and cocharacter lattices with the duality pairing (m,n)\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{\cdot}\hss}\raisebox{-1.29167pt}{\cdot}}=n(m)\in\mathbb{Z}, $m\in M$, $n\in N$.

We use notation from [2]. Let $\Delta$ be a fan in $N$ such that the associated toric variety $X=X(\Delta)$ is projective. Let $P$ be the set of rays of $\Delta$ and $v_{\rho}\in N$, $\rho\in P$, their generators, i.e. $\lvert{\rho}\rvert\cap N={\mathbb{Z}}_{\geq 0}\cdot v_{\rho}$. For any cone $\sigma\in\Delta$, let $T_{\sigma}\subset T$ denote the stabilizer of a point of the $T$-orbit in $X$ corresponding to $\sigma$. Note that $M(T_{\sigma})=M/({\sigma}^{\perp}\cap M)$. We assume that all vector spaces and bundles have finite rank.

According to Klyachko [3], a $T$-equivariant vector bundle $\mathcal{E}$ on $X$, customarily called ’a toric vector bundle’, is determined, up to an isomorphism, by a collection of complete vector space $\mathbb{Z}$-filtrations

[TABLE]

of the fiber $E={\mathcal{E}}_{o}$ over $o=(1,\ldots,1)\in T\subset X$ satisfying the following compatibility condition:

for any cone $\sigma\in\Delta$, there is a grading $E=\bigoplus_{\begin{subarray}{c}\chi\in M(T_{\sigma})\end{subarray}}E_{\chi}^{\sigma}$ such that for any $\rho\in P$, $\rho\prec\sigma$, $j\in\mathbb{Z}$

[TABLE]

One can similarly define filtrations of $E$ for any $n\in N$:

[TABLE]

Given a toric vector bundle $\mathcal{E}$ on $X$ and $e\in E\setminus\{0\}$, one can associate with them

• •

a piecewise linear function ${\phi}_{e}\colon N\otimes_{\mathbb{Z}}{\mathbb{Q}}\rightarrow{\mathbb{Q}}$ defined by

[TABLE]

• •

a convex polytope ${\Delta}_{e}\subset M\otimes_{\mathbb{Z}}{\mathbb{Q}}$ defined by the function ${\phi}_{e}$:

[TABLE]

Note that

[TABLE]

Moreover, there is a finite subset $\epsilon(X,\mathcal{E})\subset E\setminus\{0\}$ such that

[TABLE]

One may take $\epsilon(X,\mathcal{E})$ to be the ground set of the matroid constructed in [1]. Let $\overline{\epsilon}(X,\mathcal{E})=\{e\in{\epsilon}(X,\mathcal{E})\mid{\Delta}_{e}\cap M\neq\varnothing\}.$

The vector bundle $\mathcal{E}$ is big if and only if there is an integer $p>0$ such that

[TABLE]

i.e.

[TABLE]

Here $d=\dim X+\operatorname{rk}{\mathcal{E}}-1$. A convenient reference is [4], Theorem $3.3$, or work of Khovanskii.

If $p>0$, then the vector bundle $\operatorname{Sym}^{p}{\mathcal{E}}$, the fiber of which over $o$ is $\operatorname{Sym}^{p}{E}$, corresponds to the following filtrations:

[TABLE]

This implies that if $f={f}^{c_{1}}_{1}\ldots{f}^{c_{q}}_{q}\in\operatorname{Sym}^{\sum_{i}c_{i}p_{i}}{E}$, $f_{i}\in\operatorname{Sym}^{p_{i}}{E}$, $c_{i}\geq 0$, $i=1,\ldots,q$, then

[TABLE]

where $+$ denotes Minkowski addition.

2 The main result

Given a polytope $\pi\subset{\mathbb{Q}}^{r}$, let $L\pi=\operatorname*{Span}\{a-b\mid a\in\pi,b\in\pi\}\subset{\mathbb{Q}}^{r}$ be the vector subspace spanned by the differences of points of $\pi$. Similarly, the toric vector bundle $\mathcal{E}$ on $X$ defines a vector subspace

[TABLE]

If $w\colon M\otimes_{\mathbb{Z}}{\mathbb{Q}}\rightarrow M\otimes_{\mathbb{Z}}{\mathbb{Q}}/L(X,\mathcal{E})$ is the natural linear projection, then we define a subsemigroup

[TABLE]

Note that, unless $\mathcal{E}=0$, $L(X,\mathcal{E})=L{\Delta}_{e}$ for some $e\in\operatorname{Sym}^{a}{E}\setminus\{0\}$, $a>0$. Moreover, one can then find such $e$ in the ground set of the matroid constructed in [1] for $\operatorname{Sym}^{a}{\mathcal{E}}$.

For any $p>0$, consider the finite set $W_{p}\subset M\otimes_{\mathbb{Z}}{\mathbb{Q}}/L(X,\mathcal{E})$ of points

[TABLE]

Let ${\Lambda}_{p}\subset\Lambda\times\mathbb{Z}$ be the subsemigroup generated by $W_{p}\times\{1\}$ and ${\Delta}^{\perp}(p,X,\mathcal{E})\subset M\otimes_{\mathbb{Z}}{\mathbb{Q}}/L(X,\mathcal{E})$ be the convex hull of $W_{p}$. Consider the following algebra graded by ${\Lambda}_{p}$ and generated by $\overline{\epsilon}(X,\operatorname{Sym}^{p}{\mathcal{E}})=\{f_{1},\ldots,f_{q}\}$:

[TABLE]

[TABLE]

Multiplication in ${\mathcal{A}}(p,X,\mathcal{E})$ is inherited from the symmetric algebra $\operatorname{Sym}{E}$. Let

[TABLE]

Assume that the subset

[TABLE]

of the symmetric algebra is closed under multiplication and contains the ground sets of the matroids constructed in [1] for $\operatorname{Sym}^{p}{\mathcal{E}}$, $p>0$.

The main result of this note is the following criterion of bigness.

Theorem 2.1**.**

Let $X=X(\Delta)$ be a projective toric variety, $\mathcal{E}$ a toric vector bundle on $X$. Then the following are equivalent:

$\mathcal{E}$* is big,* 2. 2.

there is an integer $p>0$ such that ${\alpha}(p,X,\mathcal{E})>0$.

Proof.

We may assume that $\dim X>0$.

If $\mathcal{E}$ is big, then there is an integer $p>0$ such that (1) holds. If $\overline{\epsilon}(X,\operatorname{Sym}^{p}{\mathcal{E}})=\{f_{1},\ldots,f_{q}\}$, then (1) implies that

[TABLE]

This expression is bounded from above by

[TABLE]

and so ${\alpha}(p,X,\mathcal{E})>0$.

Suppose that there is an integer $p>0$ such that ${\alpha}(p,X,\mathcal{E})>0$. We may assume that $L(X,\mathcal{E})=L{\Delta}_{e}$ for some $e\in\overline{\epsilon}(X,\operatorname{Sym}^{p}{\mathcal{E}})=\{f_{1},\ldots,f_{q}\}$. Choose an integer $N>0$ large enough so that

[TABLE]

has a nonempty interior in $L(X,\mathcal{E})$, where $\tilde{{\Delta}}_{f_{i}}$ denotes the orthogonal projection of ${\Delta}_{f_{i}}$ onto $L(X,\mathcal{E})$ with respect to some positive definite integral symmetric bilinear form on $M$. In particular, every polytope $(1+c_{1})\cdot\tilde{{\Delta}}_{f_{1}}+\ldots+(1+c_{q})\cdot\tilde{{\Delta}}_{f_{q}}$, where $\max_{i}\lvert c_{i}\rvert\leq 1/N$, contains a fixed ball of dimension $\dim L(X,\mathcal{E})$ and radius $s>0$. Thus, if $f=(f_{1}\ldots f_{q})^{lN}\cdot f^{c_{1}}_{1}\ldots f^{c_{q}}_{q}$, where $c_{i}\geq 0$, $\sum_{i}c_{i}\cdot w_{f_{i}}=w$, $\sum_{i}c_{i}=l$, then ${\Delta}_{f}$ contains a ball $B_{w,l}$ of dimension $\dim L(X,\mathcal{E})$ and radius $lN\cdot s$, which depends on $(w,l)\in{\Lambda}_{p}$ but not on particular numbers $c_{i}$. Hence

[TABLE]

where $C>0$ does not depend on $(w,l)\in{\Lambda}_{p}$.

This implies that

[TABLE]

[TABLE]

[TABLE]

and so $\mathcal{E}$ is big by definition.

∎

Corollary 2.2**.**

Suppose $\mathcal{E}$ is a toric vector bundle on a projective toric variety $X=X(\Delta)$. If there is an integer $a>0$ and $f\in\operatorname{Sym}^{a}{E}\setminus\{0\}$ such that $\dim{\Delta}_{f}=\dim X$, then $\mathcal{E}$ is big.

Proof.

If $b>0$ is large enough, then $M\cap{\Delta}_{f^{b}\cdot g}\neq\varnothing$ for any $g\in{\epsilon}(X,\operatorname{Sym}^{a}{\mathcal{E}})$. As the subset

[TABLE]

contains a vector space basis of the symmetric algebra and $\dim L(X,\mathcal{E})=\dim X$, ${\alpha}(a+ab,X,\mathcal{E})>0$.

∎

Example 2.3**.**

Suppose $\mathcal{E}=\mathcal{O}(D_{1})\oplus\cdots\oplus\mathcal{O}(D_{t})$, $t>0$, is a split toric vector bundle on a projective toric variety $X=X(\Delta)$. Then $\mathcal{E}$ is big if and only if $\dim L(X,\mathcal{E})=\dim X$ if and only if there are nonnegative integers $a_{1},\ldots,a_{t}$ such that $\mathcal{O}(a_{1}D_{1}+\ldots+a_{t}D_{t})$ is big.

Indeed, let $\bigcup_{p>0}{\epsilon}(X,\operatorname{Sym}^{p}{\mathcal{E}})\subset\operatorname{Sym}{E}$ be the set of all nonconstant monomials in $x_{1},\ldots,x_{t}$, where $x_{1},\ldots,x_{t}\in E$ are generators of the splitting summands of $\mathcal{E}$. For any $p>0$, ${\mathcal{A}}(p,X,\mathcal{E})$ is a subalgebra of the symmetric algebra $\operatorname{Sym}{E}$:

[TABLE]

Hence ${\alpha}(p,X,\mathcal{E})=0$ unless $\dim L(X,\mathcal{E})=\dim X$.

Acknowledgement

The author is grateful to the School of Mathematics of Sun Yat-sen University for support and excellent working conditions.