A weak criterion of bigness for toric vector bundles
Evgeny Mayanskiy
math.AG



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A weak criterion of bigness for toric vector bundles
    Evgeny Mayanskiy
    December 30, 2023
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  We prove a weak version of a bigness criterion for equivariant vector bundles on toric varieties. 




§ PRELIMINARIES


Let k be an algebraically closed field of characteristic 0, T=(k^*)^r a torus, M=M(T)≅ℤ^r and N=M(T)^∨ the character and cocharacter lattices with the duality pairing (m,n) n(m)∈ℤ, m∈ M, n∈ N.


We use notation from <cit.>. Let Δ be a fan in N such that the associated toric variety X=X(Δ) is projective. Let P be the set of rays of Δ and v_ρ∈ N, ρ∈ P, their generators, i.e. |ρ|∩ N = ℤ_≥ 0· v_ρ. For any cone σ∈Δ, let T_σ⊂ T denote the stabilizer of a point of the T-orbit in X corresponding to σ. Note that M(T_σ) = M/(σ^⊥∩ M). We assume that all vector spaces and bundles have finite rank.


According to Klyachko <cit.>, a T-equivariant vector bundle ℰ on X, customarily called 'a toric vector bundle', is determined, up to an isomorphism, by a collection of complete vector space ℤ-filtrations

    ℱ_ρ (ℰ) = { E ⊇⋯⊇ℰ^ρ(j) ⊇ℰ^ρ(j+1) ⊇⋯⊇ (0)  },  ρ∈ P,

of the fiber E=ℰ_o over o=(1,… ,1)∈ T⊂ X satisfying the following compatibility condition:



for any cone σ∈Δ, there is a grading E = ⊕_χ∈ M(T_σ) E_χ^σ such that for any ρ∈ P, ρ≺σ, j∈ℤ

    ℰ^ρ(j) = ⊕_χ∈ M(T_σ) 
     (χ , v_ρ)≥ j E_χ^σ.





One can similarly define filtrations of E for any n ∈ N:

    ℰ^n(j) = ⊕_χ∈ M(T_σ) 
     (χ , n)≥ j E_χ^σ  if   n∈|σ| .

Given a toric vector bundle ℰ on X and e∈ E ∖{ 0 }, one can associate with them


  * a piecewise linear function ϕ_e N⊗_ℤℚ→ℚ defined by

    ϕ_e (n) = max_j{ j | e∈ℰ^n(j) },   n∈ N, and


  * a convex polytope Δ_e⊂ M⊗_ℤℚ defined by the function ϕ_e:

    Δ_e = { u ∈ M⊗_ℤℚ| (u,v_ρ) ≤ϕ_e (v_ρ), ρ∈ P } .



Note that 

    H^0(X,ℰ)=⊕_u∈ M*Span{χ^-u⊗ e | u∈Δ_e }⊂ k[M]⊗_kE  <cit.>.


Moreover, there is a finite subset ϵ (X,ℰ)⊂ E ∖{ 0 } such that 

    H^0(X,ℰ)=*Span_e∈ϵ (X,ℰ){χ^-u⊗ e | u∈Δ_e∩ M }.

One may take ϵ (X,ℰ) to be the ground set of the matroid constructed in <cit.>. Let ϵ(X,ℰ)={ e∈ϵ(X,ℰ) |Δ_e∩ M ≠∅} .


The vector bundle ℰ is big if and only if there is an integer p > 0 such that

    lim sup_l →∞Im (S^lH^0(X,Sym^pℰ) → H^0(X,Sym^plℰ))/(pl)^d/d! > 0,

i.e.

    lim sup_l →∞( 1/l^d·*Span_e_1,… , e_l∈ϵ (X,Sym^pℰ){χ^-(u_1 + … + u_l)⊗ (e_1 ⋯ e_l) | u_i∈Δ_e_i∩ M,  i=1,… , l }) > 0.

Here d= X + rkℰ - 1. A convenient reference is <cit.>, Theorem 3.3, or work of Khovanskii.


If p > 0, then the vector bundle Sym^pℰ, the fiber of which over o is Sym^pE,  corresponds to the following filtrations:

    ( Sym^pℰ)^ρ(j) = ∑_j_1 + … + j_p = jℰ^ρ(j_1)·…·ℰ^ρ(j_p) ⊂Sym^pE,   j∈ℤ , ρ∈ P.


This implies that if f=f^c_1_1…f^c_q_q∈Sym^∑_i c_ip_iE, f_i∈Sym^p_iE, c_i≥ 0, i=1,… ,q, then

    c_1·Δ_f_1 + … + c_q·Δ_f_q⊂Δ_f,

where + denotes Minkowski addition.



§ THE MAIN RESULT


Given a polytope π⊂ℚ^r, let Lπ = *Span{ a - b | a∈π, b∈π}⊂ℚ^r be the vector subspace spanned by the differences of points of π. Similarly, the toric vector bundle ℰ on X defines a vector subspace

    L(X,ℰ) = *Span{ a - b | a, b∈Δ_f ,  f∈Sym^pE∖{ 0 } ,  p > 0 }⊂ M⊗_ℤℚ.

If w M⊗_ℤℚ→ M⊗_ℤℚ / L(X,ℰ) is the natural linear projection, then we define a subsemigroup

    Λ = { w(Δ_f) | f∈Sym^pE∖{ 0 } ,  p > 0 , Δ_f ≠∅}⊂ M⊗_ℤℚ / L(X,ℰ).


Note that, unless ℰ = 0, L(X,ℰ) = LΔ_e for some e∈Sym^aE∖{ 0 }, a > 0. Moreover, one can then find such e in the ground set of the matroid constructed in <cit.> for Sym^aℰ.


For any p > 0, consider the finite set W_p ⊂ M⊗_ℤℚ / L(X,ℰ) of points

    w_f = w(Δ_f),   f∈ϵ (X,Sym^pℰ) .

Let Λ_p ⊂Λ×ℤ be the subsemigroup generated by W_p×{ 1 } and Δ^⊥(p,X,ℰ) ⊂ M⊗_ℤℚ / L(X,ℰ) be the convex hull of W_p. Consider the following algebra graded by Λ_p and generated by ϵ (X,Sym^pℰ)={ f_1,… ,f_q }:

    𝒜(p,X,ℰ)=⊕_w∈Λ_p𝒜_w(p,X,ℰ),


    𝒜_(w,l)(p,X,ℰ)=*Span{f^c_1_1…f^c_q_q | c_i≥ 0,  c_1· w_f_1 + … + c_q· w_f_q = w,  c_1 + … + c_q= l }⊂Sym^plE.

Multiplication in 𝒜(p,X,ℰ) is inherited from the symmetric algebra SymE. Let

    α(p,X,ℰ)=lim sup_l→∞∑_w∈Λ ,   (w,l)∈Λ_p𝒜_(w,l)(p,X,ℰ) / l^ X -  L(X,ℰ)+rkℰ - 1∈ℝ.


Assume that the subset

    ⋃_p>0ϵ(X,Sym^pℰ) ⊂⊕_p≥ 0Sym^pE = SymE

of the symmetric algebra is closed under multiplication and contains the ground sets of the matroids constructed in <cit.> for Sym^pℰ, p>0.


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


Let X=X(Δ) be a projective toric variety, ℰ a toric vector bundle on X. Then the following are equivalent:


  * ℰ is big,

  * there is an integer p > 0 such that α(p,X,ℰ) > 0.





We may assume that X > 0.


If ℰ is big, then there is an integer p>0 such that (<ref>) holds. If ϵ(X,Sym^pℰ) = { f_1,… ,f_q }, then (<ref>) implies that

    0 < lim sup_l→∞( 1/l^d·*Span_c_1+… +c_q = l 
     c_i ≥ 0{χ^-u⊗f^c_1_1⋯ f^c_q_q| u ∈( c_1 ·Δ_f_1 + … + c_q ·Δ_f_q)∩ M }).


This expression is bounded from above by

    ( lim sup_l→∞#( M ∩ l·( Δ_f_1+ … + Δ_f_q) )/l^ L(X,ℰ)) ·α(p,X,ℰ),

and so α(p,X,ℰ) > 0.


Suppose that there is an integer p>0 such that α(p,X,ℰ) > 0. We may assume that L(X,ℰ) = LΔ_e for some e∈ϵ(X,Sym^pℰ) = { f_1,… , f_q }. Choose an integer N > 0 large enough so that

    ⋂_c_1,… , c_q∈ℝ
    max_i | c_i |≤ 1/N (1+c_1)·Δ̃_f_1 + … + (1+c_q)·Δ̃_f_q

has a nonempty interior in L(X,ℰ), where Δ̃_f_i denotes the orthogonal projection of Δ_f_i onto L(X,ℰ) with respect to some positive definite integral symmetric bilinear form on M. In particular, every polytope (1+c_1)·Δ̃_f_1 + … + (1+c_q)·Δ̃_f_q, where max_i | c_i |≤ 1/N, contains a fixed ball of dimension L(X,ℰ) and radius s > 0. Thus, if f=(f_1… f_q)^lN· f^c_1_1… f^c_q_q, where c_i≥ 0, ∑_i c_i· w_f_i = w, ∑_i c_i=l, then Δ_f contains a ball B_w,l of dimension L(X,ℰ) and radius lN· s, which depends on (w,l)∈Λ_p but not on particular numbers c_i. Hence

    #( M∩⋂_f=(f_1… f_q)^lN· f^c_1_1… f^c_q_q 
     c_i≥ 0,  ∑_i c_i· w_f_i = w,  ∑_i c_i=lΔ_f ) ≥ C· l^ L(X,ℰ),

where C > 0 does not depend on (w,l)∈Λ_p.


This implies that

    lim sup_l→∞ H^0(X,Sym^p(Nql+l)ℰ)/(Nql+l)^d≥


    ≥lim sup_l→∞( 1/((Nq+1)l)^d·*Span_c_1+… +c_q = l 
     c_i≥ 0{χ^-u⊗ f | u ∈Δ_f∩ M,  f=(f_1… f_q)^lN· f^c_1_1… f^c_q_q })≥



    ≥C/(Nq+1)^d·α(p,X,ℰ) > 0,

and so ℰ is big by definition.




Suppose ℰ is a toric vector bundle on a projective toric variety X=X(Δ). If there is an integer a>0 and f∈Sym^aE∖{ 0 } such that Δ_f =  X, then ℰ is big.




If b>0 is large enough, then M∩Δ_f^b· g≠∅ for any g∈ϵ (X,Sym^aℰ). As the subset 

    ⋃_p>0ϵ(X,Sym^pℰ) ⊂SymE

contains a vector space basis of the symmetric algebra and L(X,ℰ) =  X, α(a+ab,X,ℰ) > 0.




Suppose ℰ = 𝒪 (D_1) ⊕⋯⊕𝒪 (D_t), t>0, is a split toric vector bundle on a projective toric variety X=X(Δ). Then ℰ is big if and only if L(X,ℰ) =  X if and only if there are nonnegative integers a_1,… , a_t such that 𝒪 (a_1D_1+… + a_tD_t) is big.


Indeed, let ⋃_p>0ϵ(X,Sym^pℰ) ⊂SymE be the set of all nonconstant monomials in x_1,… ,x_t, where x_1,… , x_t∈ E are generators of the splitting summands of ℰ. For any p>0, 𝒜(p,X,ℰ) is a subalgebra of the symmetric algebra SymE:

    ⊕_w∈Λ , (w,l)∈Λ_p𝒜_(w,l)(p,X,ℰ)⊂Sym^plE,   l>0.

Hence α(p,X,ℰ) = 0 unless L(X,ℰ) =  X.




§ ACKNOWLEDGEMENT


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



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