Seshadri constants of parabolic vector bundles

Let $X$ be a complex projective variety, and let $E_{\ast}$ be a parabolic vector bundle on $X$. We introduce the notion of \textit{parabolic Seshadri constants} of $E_{\ast}$. It is shown that these constants are analogous to the classical Seshadri constants of vector bundles, in particular, they have parallel definitions and properties. We prove a Seshadri criterion for parabolic ampleness of $E_{\ast}$ in terms of parabolic Seshadri constants. We also compute parabolic Seshadri constants for symmetric powers and tensor products of parabolic vector bundles.


Introduction
Let X be a complex projective variety, and let L be a nef line bundle on X.For a point x ∈ X, the Seshadri constant of L at x, denoted by ε(X, L, x), is defined to be ε(X, L, x) where the infimum is taken over all irreducible and reduced curves C ⊂ X passing through x; here L • C denotes the intersection number (equivalently, the degree of L restricted to C), while mult x C denotes the multiplicity of the curve C at x.Alternatively, we can define Seshadri constants as follows.Let π : X −→ X be the blow up of X at x, and let A denote the corresponding exceptional divisor.Then ε(X, L, x) = sup {λ ≥ 0 π * (L) − λA is nef}.
This constant was first introduced by Demailly in [Dem] while he was studying problems related to generation by jets and Fujita's Conjecture.The name originated from an ampleness criterion of Seshadri [Har2,Theorem I.7.1].Seshadri constant of line bundles blossomed into a very active area of research, especially in connection with the positivity properties of the line bundle L. Seshadri constants for vector bundles of arbitrary rank have also been studied.They were defined in [Hac] and studied in depth in [FM].See [HMP,BHN,DKS,MR2] for some results on Seshadri constants of vector bundles.A parabolic vector bundle on a curve is a vector bundle E with some additional structures given by filtrations of certain fibers of E along with weights attached to these filtrations (see Definition 2.1).We recall that parabolic vector bundles were introduced by Seshadri (see [Se]) while studying certain representations of the fundamental groups of punctured Riemann surfaces; they were further studied by Mehta and Seshadri in [MS].The notion of parabolic vector bundle was generalized to higher dimensions by Maruyama and Yokogawa in [MY].
There has been a lot of research on parabolic vector bundles, starting with a description of their moduli spaces in [MS, MY].In [Bi2], certain orbifold bundles were associated to parabolic bundles (see also [Bo1], [Bo2]).The notion of a ramified principal GL n (C)-bundle, which generalizes the notion of principal GL n (C)-bundles, was introduced in [BBN].It was shown there that parabolic vector bundles can be viewed as ramified GL n (C)-bundles.Using this identification, [BL] constructed a projectivization of a parabolic vector bundle as well as the tautological line bundle on the projectivization.Many positivity properties, such as ampleness of line bundles, have been generalized to vector bundles.Some of these notions have also been generalized to the context of parabolic vector bundles.For example, the notion of ample parabolic bundles was defined in [Bi1].The ampleness of a parabolic vector bundle is characterized by the ampleness of the tautological bundle on the projectivization associated to that parabolic bundle (see [BL]).
Proceeding further in this direction, it is natural to ask whether other positivity notions can also be generalized to the set-up of parabolic vector bundles.As remarked earlier, Seshadri constant is an important invariant in the study of positivity.In this paper, we generalize the notion of Seshadri constants to parabolic vector bundles on projective varieties.Crucial ingredients in our construction are orbifold bundles associated to the parabolic bundles and projectivization of parabolic bundles.Many of our results and proofs are motivated by analogous results of Fulger and Murayama in [FM].
In Section 2, we recall the definition and basic properties of parabolic vector bundles that we use.They are included for the sake of completeness and also for the convenience of the reader.
Let X be an irreducible projective variety and let E * be a parabolic vector bundle on X.In Section 3, we give the definition of the parabolic Seshadri constant ε * (E * , x) of E * at any point x ∈ X; see Definition 3.1.This definition makes use of the notion of projectivization of a parabolic bundle; see Section 2.4 for a description of this construction.Alternately, one can also use the notion of orbifold bundles (described in Section 2.3) to define parabolic Seshadri constants.
Different alternative characterizations of parabolic Seshadri constants are given in Theorem 3.3 and Theorem 3.5.
We also prove an analogue of the Seshadri criterion for ampleness for parabolic bundles (see Theorem 3.6).We state this criterion here for the convenience of the reader.See Section 3 for the notation.
Theorem (Theorem 3.6).Let E * be a parabolic vector bundle on a smooth irreducible projective variety X such that the numerical class where the infimum is taken over all points of X.
An upper bound for parabolic Seshadri constants is given in Theorem 3.7.
In Section 4, we prove several properties of parabolic Seshadri constants.We define the notion of multi-point Seshadri constants for the usual vector bundles and relate it to the parabolic Seshadri constants.
One of the results proved in this section describes how parabolic Seshadri constants can be computed using restriction to curves.
Theorem (Theorem 4.3).Let E * be a parabolic nef vector bundle on a smooth irreducible complex projective variety X, and let E ′ −→ Y be the corresponding orbifold bundle over Y (see Section 2.3).Then where the infimum is taken over all irreducible curves C ⊂ Y such that C γ −1 (x) = ∅, and ν : C −→ C is the normalization map.
We also compute parabolic Seshadri constants of symmetric powers of parabolic vector bundles (see Theorem 4.5) as well as tensor products of parabolic vector bundles (see Theorem 4.6).These results are stated below.
Theorem (Theorem 4.5).Let E * be a parabolic nef vector bundle on a smooth irreducible complex projective variety X.Then for any positive integer for every point x ∈ X.
Theorem (Theorem 4.6).Let E * and F * be two parabolic nef vector bundles on a smooth irreducible complex projective variety X having a common parabolic divisor D ⊂ X.Then for every point Some examples and questions are mentioned in Section 4.1.We work throughout over the field C of complex numbers.The field of real numbers is denoted by R.

Preliminaries
In this section, we briefly recall the definitions, properties and constructions associated to parabolic vector bundles; more details can be found in [Bi1,Bi2,Bi3,BS,BBN,BL].
Definition 2.1.Let X be a connected smooth complex projective variety of dimension d, and let D ⊂ X be an effective divisor on X.
(1) Let E be a torsion-free coherent O X -module.A quasi-parabolic structure on E with respect to D is a filtration by O X -coherent subsheaves where The above integer l is called the length of the filtration.(2) A parabolic structure on E with respect to D is a quasi-parabolic structure as above together with a system of weights The numbers {α i } l i=1 are called parabolic weights and we say that α i is attached to F i (E).
(3) A parabolic sheaf is a torsion-free coherent O X -module E with a parabolic structure.
We denote a parabolic sheaf by the triple E, F * , α * , where F * denotes the above filtration and α * denotes the system of parabolic weights.When F * and α * are clear from the context, we denote E, F * , α * by E * .(4) If E is a vector bundle on X, then E * is called a parabolic vector bundle with parabolic divisor D. The rank of a parabolic vector bundle is simply the rank of the underlying vector bundle E.
For any parabolic sheaf E * defined as above, and for any t ∈ R, define the following filtration {E t } t∈R of coherent sheaves parametrized by R: where [t] is the integral part of t and α i−1 < t − [t] ≤ α i with α 0 = α l − 1 and α l+1 = 1.The filtration {E t } t∈R evidently determines the parabolic structure E, F * , α * uniquely.Note that any coherent subsheaf M ⊂ E has an induced parabolic structure such that the corresponding filtration {M t } t∈R is defined by Consider the decomposition where every D j is a reduced irreducible divisor, and b j ≥ 1.Let denote the inclusion map of the subscheme b j D j .For each 1 ≤ j ≤ n, choose a filtration For every 1 ≤ j ≤ n and 1 ≤ k ≤ l j + 1, define the coherent subsheaf F i j ⊂ E using the following short exact sequences: (2. 3) The 2.2.Semistability of parabolic bundles.We fix an ample line bundle L on X.For any coherent O X -module E, the degree of E with respect to L is defined to be (see [MY]).The parabolic degree of E * with respect to L, denoted by par deg(E * ), is defined as follows: The parabolic slope of E * , denoted by par µ(E * ), is the quotient par deg (E * )   rank(E) ∈ R.
Assumption 2.5.Henceforth we will always impose the following four conditions on the parabolic bundles E * , with parabolic divisor D, that we will consider: 2) are subbundles of f * i E for every i.(c) All the weights α i j are rational numbers; so α i j = m i j /N, where N is some fixed integer and and a decomposition of E Ux into a direct sum of line bundles, such that the filtration of all E Ux∩D i , 1 ≤ i ≤ n, are constructed using the decomposition.To explain this condition in detail, let ).The condition says that there is a holomorphic decomposition into a direct sum of line bundles where r E = rank(E), such that every F k j Ux∩D k considered above is of the form where r k,j = rank(F k j ) and 1 2.3.Orbifold bundles.Let Y be a smooth complex projective variety; its group of algebraic automorphisms will be denoted by Aut(Y ).Let Γ be a finite group and ψ : Γ −→ Aut(Y ) a homomorphism giving an action of Γ on Y .
Definition 2.6.An orbifold bundle on Y , with Γ as the orbifold group, is a vector bundle V on Y together with a lift of the action of Γ on Y to V , i.e., Γ acts on the total space of V such that the action of any g ∈ Γ gives a vector bundle isomorphism between V and ψ(g −1 ) * V .A subsheaf F of an orbifold bundle V is called an orbifold subsheaf if the action of Γ on V preserves F .A homomorphism V −→ V ′ between orbifold bundles is a homomorphism V −→ V ′ between the vector bundles that commutes with the actions of Γ on V and V ′ .
Let L be an orbifold line bundle on Y which is also ample.Then we can define the degree of any coherent sheaf on Y using L as done in (2.5).
Definition 2.7.An orbifold bundle V on Y is called orbifold semistable (respectively, orbifold stable) if for any orbifold subsheaf F of V with 0 < rank(F ) < rank(E) the following inequality holds: Now let X be a smooth projective complex variety, and let D be an effective divisor on X satisfying Assumption 2.5(a).Fix an integer N ≥ 2. From the Covering Lemma of Y. Kawamata, [Ka,Theorem 17] and [KMM,Theorem 1.1.1],we know that there is a connected smooth projective variety Y and a (ramified) Galois covering morphism where k i are some positive integers.Let E * be a parabolic bundle on X.Then using the Galois cover γ : Y −→ X in (2.6) we can construct an orbifold bundle V on Y such that the parabolic bundle E * is recovered from it by taking Γ-invariants of the direct image of the twists of V using the irreducible components of D (see [Bi2], [Bo1], [Bo2] for the explicit construction).Note that the same γ is used for all parabolic bundles E * with the same parabolic divisor D.
For the convenience of the reader we will briefly recall the construction of an orbifold bundle on Y from a parabolic bundle on X with parabolic structure on D. First let L * be a parabolic line bundle on X with parabolic structure on D. So L is a line bundle on X, and for each 1 ≤ i ≤ n we have 0 ≤ α i = m i 1 N < 1 which is the parabolic weight over the component D i of D; see Assumption 2.5(c).Then the orbifold line bundle L ′ on Y corresponding to the parabolic line bundle L * is (see (2.7) for k i ); note that each (γ * D i ) red is preserved by the action of Γ on Y and hence the above line bundle has a natural orbifold structure.The orbifold structure on L ′ is given by the orbifold structures on γ * L and Note that this construction is local in the sense that we do not need X or Y to be a projective variety.Let S ⊂ D denote the singular locus of D, and let be the inclusion map.For a parabolic bundle E * on X, consider the restriction of it to the complement X \ S. It can be locally expressed as direct sums of parabolic line bundles.Therefore, using the above construction of an orbifold line bundle from a parabolic line bundle, and patching locally defined orbifold bundles, we get an orbifold vector bundle E on Y \ γ −1 (S).For the patching to work compatibly, note that if V 1 and V 2 are two orbifold bundles on γ −1 (U) for two decompositions of the parabolic bundle E * U into direct sum of parabolic line bundles, then the natural isomorphism of orbifold bundles , where ι is the map in (2.8).The conditions in Assumption 2.5 ensure that the coherent sheaf ι * E is locally free.The action of Γ on E produces an action of Γ on ι * E. This orbifold bundle ι * E is the one associated to the parabolic bundle E * .
Let L be an ample line bundle on X.Consider L := γ * L which is an orbifold line bundle on Y .Since γ is a finite morphism, and L is ample, the line bundle L is also ample.We will use L (respectively, L) to define the degree of a coherent sheaf on X (respectively, Y ).
The above mentioned correspondence between parabolic bundles and orbifold bundles satisfies the following compatibility condition with semistability.
Proposition 2.8 ([Bi2, Lemma 2.7, Lemma 3.16]).The above orbifold bundle V corresponding to the parabolic bundle E * is orbifold semistable with respect to L if and only if E * is parabolic semistable with respect to L. Also, V is semistable with respect to L in the usual sense if and only if E * is parabolic semistable with respect to L. Proposition 2.9 ([Bi1, Lemma 4.6], [Bi3,Lemma 2.18], [BS,Proposition 3.2]).A parabolic bundle E * is parabolic ample (respectively, nef ) if and only if the corresponding orbifold bundle V is ample (respectively, nef ) as a vector bundle (in the usual sense).
Parabolic Chern classes, denoted by c i (E * ), of a parabolic bundle E * have been introduced in [Bi3] (see also [IS]).Let V be the associated orbifold bundle on Y corresponding to a parabolic bundle E * (satisfying Assumption 2.5) for a suitable Galois covering map γ : Y −→ X.Then for all i ≥ 0 (see [Bi3,Section 3]).
2.4.Parabolic bundles as ramified GL(n, C)-bundles.Let X be a smooth complex projective variety, and let D be a normal crossing divisor on X.Let φ : E GL(n,C) −→ X be a ramified principal GL(n, C)-bundle over X with ramification over D. This means the total space E GL(n,C) is a smooth complex quasiprojective variety equipped with an algebraic right action of GL(n, C) satisfying the following five conditions: (1) (2) for each point x ∈ X, the action of GL(n, C) on the reduced fiber φ −1 (x) red is transitive, (3) the restriction of φ to φ −1 (X − D) a principal GL(n, C)-bundle over X − D, (4) for each irreducible component D i ⊂ D, the reduced inverse image φ −1 (D i ) red is a smooth divisor and is a normal crossing divisor on E GL(n,C) , and (5) for any point x ∈ D, and any point z ∈ φ −1 (x), the isotropy subgroup G z ⊂ GL(n, C), for the action of GL(n, C) on E GL(n,C) , is a finite group, and if x is a smooth point of D, then the natural action of G z on the quotient line Note that the last condition implies that the isotropy subgroup G z ⊂ GL(n, C) is actually a finite cyclic group.
There is a natural bijective correspondence between the complex vector bundles of rank n on X and the principal GL(n, C)-bundles on X.This bijection sends a principal GL(n, C)-bundle F to the vector bundle F × GL(n,C) C n associated to F for the standard action of GL(n, C) on C n .This correspondence extends to a bijective correspondence between the ramified principal GL(n, C)bundles with ramification over D and parabolic vector bundles of rank n with D as the parabolic divisor (see [BBN,Theorem 1.1]).We next recall the construction of projectivization of a parabolic bundle.Let E * be a parabolic vector bundle over X of rank n.Let φ : E GL(n,C) −→ X be the corresponding ramified principal GL(n, C)-bundle with ramification divisor D. Consider the standard action of GL(n, C) on C n ; it induces an action of GL(n, C) on the projective space P n−1 .The projectivization of E * , denoted by P(E * ), is defined to be the associated (ramified) fiber bundle Take any point x ∈ D and any z ∈ φ −1 (x).Let G z ⊂ GL(n, C) be the isotropy subgroup for z for the action of GL(n, C) on E GL(n,C) .We recall that G z is a finite group (by condition (5) above).Let n x be the order of G z .Note that the order of the group G z is independent of the choice of z ∈ φ −1 (x) because GL(n, C) acts transitively on φ −1 (x).The number of distinct integers n x as x varies over D is finite (see [BL]).Let (2.9) be the least common multiple of all these finitely many positive integers n x .For any point y ∈ P n−1 , let H y ⊂ GL(n, C) be the isotropy subgroup for the natural action of GL(n, C) on P n−1 ; so H y is a maximal parabolic subgroup of GL(n, C).The group H y then acts on the fiber of O P n−1 (1) −→ P n−1 over the point y.From the definition of N(E * ) in (2.9) it follows immediately that for any z ∈ φ −1 (D) and any y ∈ P n−1 , the subgroup G z H y ⊂ GL(n, C) acts trivially on the fiber of the line bundle O P n−1 N(E * ) := O P n−1 (1) ⊗N (E * ) over the point y.
Consider the action of GL(n, C) on the total space of O P n−1 N(E * ) constructed using the standard action of GL(n, C) on C n .Let be the associated fiber bundle.As the natural projection O P n−1 N(E * ) −→ P n−1 intertwines the actions of GL(n, C) on O P n−1 N(E * ) and P n−1 , it produces a projection (2.10) Using the above observation that G z H y acts trivially on the fibers of O P n−1 N(E * ) over y it follows easily that the projection in (2.10) makes E GL(n,C) O P n−1 N(E * ) an algebraic line bundle over the projectivization P(E * ).We denote this line bundle in (2.10) by O P(E * ) (1).Let E ′ −→ Y be the orbifold bundle over Y corresponding to the parabolic vector bundle E * , where γ : Y −→ X is a covering as in (2.6) with Galois group Γ = Gal(γ).Then the action of Γ on E ′ produces a left action of Γ on P(E ′ ).Clearly, we have P(E ′ )/Γ = P(E * ).The isotropy subgroups, for the action of Γ on P(E ′ ), act trivially on the corresponding fibers of O P(E ′ ) N(E * ) ; this follows from condition (5) above in the definition of a ramified principal GL(n, C)-bundle and (2.9).Hence the quotient space O P(E ′ ) N(E * ) /Γ is actually a line bundle over P(E ′ )/Γ = P(E * ).We have a natural isomorphism of line bundles Note that the pullback of O P(E * ) (1) to P(E ′ ) under the finite quotient map ) .The parabolic bundle E * is parabolic ample (respectively, parabolic nef) if and only if the line bundle O P(E * ) (1) on P(E * ) is ample (respectively, nef) [BL,Proposition 3.3].Also, the following three statements are equivalent: • the parabolic vector bundle E * is nef, • E ′ is nef, and

Definition of Seshadri constant of parabolic bundles
Let E * be a parabolic nef vector bundle of rank n on a smooth projective variety X.We fix a point x ∈ X, and let ψ x : Bl x (X) −→ X be the blow up of X at x with exceptional divisor A x = ψ −1 x (x).Consider the following fiber product diagram: The above morphism ρ is the projectivization of the parabolic bundle E * as constructed in Section 2.4.Hence the map ρ is flat and the fiber of ρ is a P n−1 (see [BL]).Let ξ be the numerical equivalence class of the line bundle O P(E * ) (1) −→ P(E * ).
Definition 3.1.Let E * be a parabolic nef vector bundle on a smooth projective variety X.The parabolic Seshadri constant of E * at a point x ∈ X, denoted by ε * (E * , x), is defined to be Remark 3.2.As mentioned before, there is a canonical bijection between the parabolic vector bundles of rank n with parabolic divisor D and the ramified principal GL(n, C)-bundles ramified over D. In particular, the constructions of the projectivization P(E * ) and the tautological bundle O P(E * ) (1) for a parabolic bundle E * are uniquely determined by E * .So the parabolic Seshadri constant ε * (E * , x) is well-defined.
The next theorem shows that the parabolic Seshadri constant ε * (E * , x) can be computed by examining the intersection of ξ with certain curves on P(E * ).
Theorem 3.3.Let E * be a parabolic nef vector bundle on a smooth complex projective variety X, and let x ∈ X be a point of X.Let C ρ,x be the set of all integral curves C ⊂ P(E * ) that intersect the fiber ρ −1 (x) while not being contained in ρ −1 (x).Then Proof.First note that C ∈ C ρ,x if and only if mult x ρ * C > 0. Let C be the strict transform of a curve C ∈ C ρ,x under the map ψ x in (3.1).We have Since E * is given to be parabolic nef, the line bundle ξ is nef.Therefore, we conclude that for all λ ≥ 0. Every irreducible curve on Bl ρ −1 (x) P(E * ) satisfies one of the following conditions: • It is the strict transform of a curve C ∈ C ρ,x .
• It is contained in the exceptional locus ρ −1 (A x ) of ψ x .
Let y ∈ P(E * ).We can relate the classical Seshadri constant ε(ξ, y) of ξ and the parabolic Seshadri constant ε * (E * , ρ(y)) of E * at ρ(y) as follows: Hence we have ε(ξ, y) ≥ min inf Now we give another alternative characterization of parabolic Seshadri constants.This characterization uses the Galois covering γ : Y −→ X and the orbifold bundle E ′ over Y associated to E * ; see Section 2.3.Theorem 3.5.Let E * be a parabolic nef vector bundle on a smooth irreducible complex projective variety X, and let E ′ −→ Y be the corresponding orbifold bundle over Y , where γ : Y −→ X is a covering as in (2.6) with Galois group Γ = Gal(γ).Let be the blow up of Y along γ −1 (x).Consider the following fiber product diagram: Denote the numerical equivalence class of the line bundle O P(E ′ ) (1) by ξ ′ .Let A γ −1 (x) be the exceptional divisor of the map φ x .Then and mult y τ * C > 0, while N(E * ) is defined as in (2.9).
Proof.Note that . Also, we have the following commutative diagram: The universal property of a fiber product furnishes the following commutative diagram: Recall that γ ′ * (ξ) = N(E * )ξ ′ .Hence for any λ > 0, the following three statements are equivalent: To prove (3.2), let C be the strict transform of a curve C ∈ C τ,γ −1 (x) under the blow up map If C ′ is an irreducible curve in Bl τ −1 (γ −1 (x)) P(E ′ ), and it is contained in the exceptional locus Any irreducible curve on Bl τ −1 (γ −1 (x)) P(E ′ ) satisfies one of the following three conditions: Now using the nefness of φ x * (ξ ′ ) we conclude that (3.2) holds.This also completes the proof of the theorem.
We use the notation of Theorem 3.5 in what follows.
Theorem 3.6 (Seshadri criterion for parabolic ampleness).Let E * be a parabolic vector bundle on a smooth irreducible projective variety X such that the numerical class where the infimum is taken over all points of X.
Proof.Suppose that E * is parabolic ample.Then the numerical equivalence class of the tautological line bundle ξ = [O P(E * ) (1)] is ample on P(E * ).Let h be an ample divisor class on X such that ξ − δρ * h is ample for a sufficiently small δ > 0. Now for all C ∈ C ρ,x , we have The last inequality follows using the usual Seshadri criterion for ample divisors, because h is ample; see [Laz,Theorem 1.4.13].Hence inf Consequently, from Theorem 3.3 it is deduced that The last inequality again follows from the usual Seshadri criterion applied to the ample divisor h.
To prove the converse, assume that inf x∈X ε * (E * , x) > 0 for all x ∈ X.Therefore, using Theorem 3.5 we have inf Assume that E * is not parabolic ample, i.e., ξ ′ ≡ O P(E ′ ) (N(E * )) is not ample.Note that as ξ ′ is τ -ample, we have ξ ′ is nef, i.e., E * is parabolic nef.Then by the Seshadri criterion for ample divisors, one has inf So for each m ∈ N, there exist points z m ∈ P(E ′ ) and irreducible curves We claim that the curve C m is not contracted by τ for infinitely many m.If the claim is not true, C m are contracted by τ for all m ≫ 0. Then for all m ≫ 0 inf Choose an ample divisor h on Y so that ξ ′ + τ * h is ample.Hence by the Seshadri ampleness criterion for ample divisors, we conclude that inf which in fact contradicts (3.4).This proves our claim. Therefore This contradicts (3.3).Thus E * is parabolic ample.
The next theorem gives an upper bound for the parabolic Seshadri constants.Theorem 3.7.Let E * be a parabolic nef vector bundle on X, and x ∈ X a point.Then as W ranges through the subvarieties of P(E ′ ) that meet τ −1 (γ −1 (x)) without being contained in τ −1 (γ −1 (x)), where τ and γ are as in Theorem 3.5.In the above inequality, Proof.Consider the following commutative diagram where τ is the projectivization of the orbifold bundle E ′ over Y , and φ x is the blow up of Y at γ −1 (x).Recall from Theorem 3.5 that Let W ′ ⊂ P(φ * x E ′ ) be the strict transform of W by the blow up morphism φ x .Then by the above observation we have We now specialize to the situation where Note that the self-intersection number A k γ −1 (x) = 0 for k > n.Expanding binomially we thus get is ample.
Hence using the projection formula we conclude that Now we have the following equalities: From (3.5) it follows that Rearranging the terms we obtain the required bound on ε * (E * , x).
For a parabolic vector bundle E * , we define µ par min (E * ) to be the parabolic slope of the minimal parabolic semistable subquotient of E * , or in other words, µ par min (E * ) is the parabolic slope of the final piece of the graded object for the Harder-Narasimhan filtration of E * .Note that if E ′ is the orbifold bundle on Y corresponding to E * for the Galois morphism γ : Y −→ X as in (2.6), then we have Theorem 3.8.Let E * be a parabolic ample vector bundle over a smooth irreducible projective curve C with parabolic divisor D. Then for any point x ∈ C, the parabolic Seshadri constant satisfies the following: rank(E) for every point x ∈ C.
Then the cardinality of each fiber is Since C is a curve, the blow up map φ x in Theorem 3.5 is the identity morphism, and hence φ x is also the identity morphism.Consequently, Denote the numerical class of fiber of τ by f .Then where y ∈ C ′ .From [Hac,Theorem 3.1], it follows that ε(E ′ , y) = µ min (E ′ ).Using [Bi3, Equation 2.15], we also have µ min (E ′ ) = Γ µ par min (E * ).Hence the theorem follows.
Remark 3.9.If E * is a parabolic nef bundle but it is not ample, then the corresponding orbifold bundle E ′ is also nef but not ample.In this case

Properties of parabolic Seshadri constants
In this section we define the notion of multipoint Seshadri constants for nef vector bundles on irreducible projective varieties, generalizing the case of a single point.We will then give a description of the parabolic Seshadri constants using restriction to curves.Definition 4.1.Let E be a vector bundle on an irreducible complex projective variety Y , and let Consider the following commutative diagram: Let ξ and ξ ′ be the tautological bundles on P(E) and P(φ * E) respectively, and let for each 1 ≤ i ≤ n.Then the multipoint Seshadri constant of E at Z, denoted by ε(E, Z), is defined as follows: It is straight-forward to verify the following equivalent formulation: where C τ,Z is the set of all irreducible curves C ⊂ P(E) such that mult z i τ * C > 0, and Note that when Z is a singleton, the above definition recovers the usual single point Seshadri constant of a vector bundle.The single point Seshadri constants of vector bundles over smooth curves have a particularly nice description.For a nef vector bundle E over a smooth curve Y , for any point y ∈ Y , we have where C τ,y is the set of curves C ⊂ P(E) such that C intersects the fiber τ −1 (y) but it is not contained in τ −1 (y).Let B ′ ⊂ P(ν * E) (respectively, B ⊂ P(E)) be a curve which is not in any fiber of η (respectively, η).Suppose that B ′ is the strict transform of B under the blow up map ν.Then Hence for any point y ∈ C, we have .
The next result describes how parabolic Seshadri constants can be computed using restriction to curves.
Theorem 4.3.Let E * be a parabolic nef vector bundle on a smooth irreducible complex projective variety X, and let E ′ −→ Y be the corresponding orbifold bundle over Y (see Section 2.3).Then where the infimum is taken over all irreducible curves C ⊂ Y such that C γ −1 (x) = ∅, and ν : C −→ C is the normalization map.
Proof.Let C ⊂ Y be an irreducible curve such that C γ −1 (x) = ∅.Consider the following diagram: Recall that C τ,γ −1 (x) is the set of all irreducible curves W in P(E ′ ) such that W τ −1 (γ −1 (x)) and Similarly, we denote by C τc,γ −1 (x) the set of all irreducible curves where the union is taken over all irreducible curves C in Y such that C γ −1 (x) = ∅.From Theorem 3.5 we have Using (4.1) we have where the infimum is taken over all irreducible curves C in Y such that C γ −1 (x) = ∅.Now observe that for an irreducible curve C ⊂ Y with C γ −1 (x) = ∅, using Remark 4.2 it follows that inf This completes the proof.
For a parabolic vector bundle E * of rank r we define its parabolic discriminant, denoted by △ par (E * ), as follows: Remark 4.4.Let E * be a parabolic nef vector bundle on X.Then, by Theorem 4.3, for any point where the infimum is taken over all irreducible curves C ⊂ Y such that C γ −1 (x) = ∅, and ν : C −→ C is the normalization map.Note that Here ε det(E ′ ), γ −1 (x) denotes the multipoint Seshadri constants of E ′ at γ −1 (x) ⊂ Y .Note that the rank of the parabolic vector bundle E * is simply the rank of the underlying vector bundle E.Moreover, if E * is a semistable parabolic ample vector bundle with △ par (E * ) = 0 (see (4.2)), then E ′ is also semistable ample bundle on Y with vanishing discriminant.Hence in this case [BB,Theorem 1.2]).Therefore, in this case we get the following equality: for every point x ∈ X.
We now describe the parabolic Seshadri constants of symmetric powers and tensor products of parabolic vector bundles.
Theorem 4.5.Let E * be a parabolic nef vector bundle on a smooth irreducible complex projective variety X.Then for any positive integer m ε * (S m (E * ), x) = mε * (E * , x) for every point x ∈ X.
Proof.Let E ′ −→ Y be the corresponding orbifold bundle over Y as in Section 2.3.For any positive integer m, the parabolic bundle corresponding to the orbifold bundle S m (E ′ ) is the parabolic symmetric product S m (E * ) (see [Bi1]).Therefore, from Proposition 2.9 it follows that S where the infimum is taken over all irreducible curves C ⊂ Y such that C γ −1 (x) = ∅, and ν : C −→ C is the normalization map.
As taking symmetric powers commutes with the operation of pullback, and for a vector bundle V on a smooth complex projective curve C, we have This completes the proof.
Theorem 4.6.Let E * and F * be two parabolic nef vector bundles on a smooth irreducible complex projective variety X having a common parabolic divisor D ⊂ X.Then for every point x ∈ X, Proof.The correspondence between parabolic vector bundles and orbifold bundles takes tensor product of two orbifold bundles to the tensor product of the corresponding parabolic vector bundles (see [Bi1]).Let E ′ and F ′ be the orbifold bundles on Y corresponding to E * and F * respectively, where Y is the Galois cover of X as in Section 2.3.Then E ′ ⊗ F ′ corresponds to the parabolic bundle E * ⊗ F * .By Theorem 4.3, for any point x ∈ X, we have where the infimum is taken over all irreducible curves C ⊂ Y such that C γ −1 (x) = ∅, and ν : C −→ C is the normalization map (see Theorem 4.3).For the smooth curve C, we also have Therefore, This completes the proof.

4.1.
Examples and further questions.Let X be a smooth complex projective variety.Let E * be a parabolic vector bundle on X with E as the underlying vector bundle.If X has dimension one, and E is ample, then it follows that E * is parabolic ample; see [Bi1,Theorem 3.1].But in general this is not the case.We will give two examples to show that the parabolic ampleness of E * does not imply the ampleness of E and vice versa.
Let D be an effective divisor on X and suppose that D = n i=1 D i is the decomposition of D into irreducible components.A parabolic line bundle on X with parabolic divisor D is a pair L * = L, {α 1 , • • • , α n } , where L is a line bundle on X and the parabolic weight 0 ≤ α i < 1 corresponds to the divisor D i .Assume that α i ∈ Q for all i.Then L * is parabolic ample if and only if lies in the ample cone of X.Let L be a line bundle on P C (W ) which is numerically equivalent to 1 2 σ − 1 2 f .Then L is not ample by [Har3,Proposition 2.21].Next consider the parabolic bundle L * = (L, {α 1 , α 2 }) on P C (W ) with parabolic weights α 1 = 1 2 and α 2 = 1 2 .Note that c 1 (L * ) = c 1 (L) + α 1 σ + α 2 f = σ which is inside the ample cone of P C (W ).Thus L * is parabolic ample, but L is not ample.
Example 4.8.Let π : P C (W ) −→ C be a ruled surface over P 1 C defined by the normalized rank 2 bundle W = O P 1 C ⊕ O P 1 C (−1).Then the associated invariant e is equal to 1.As done above, we consider the simple normal crossing divisor σ + f , where σ denotes the normalized section of π such that O P(W ) (σ) ≃ O P(W ) (1) and f denotes a fiber of the map π.
In view of Example 4.7 and Example 4.8, using the Seshadri criterion for parabolic ampleness (see Theorem 3.6) we see that there can not be an inequality between ε * (L * , x) and ε(L, x), in general.It is still interesting to ask the following question.Question 4.9.Suppose E * is an ample parabolic bundle on a projective variety X such that the underlying vector bundle E is also ample.Then can we compare ε * (E * , x) and ε(E, x)?
Example 4.10.Let Z be a smooth complex projective variety, and let W be a vector bundle on Z.We may choose L * in such a way that c 1 (E * ) = c 1 (ρ * F * ) + rc 1 (L * ) lies in the ample cone of X.This way we can produce parabolic ample bundles on X with △ par (E * ) = 0 (see [MR1,Example 22]).Using Remark 4.4 we can compute the parabolic Seshadri constants for such parabolic bundles.
In this situation, there is the following result on the relationship between the parabolic ampleness of E * and the ampleness of its first parabolic Chern class.
Theorem 4.11 ([MR1,Theorem 19]).Let E * be a semistable parabolic vector bundle of rank r on a smooth complex projective variety X such that △ par (E * ) = 0. Then E * is parabolic ample if and only if its parabolic first Chern class c 1 (E * ) is in the ample cone of X.
So one can ask the following question.
i is a normal crossing divisor, i.e., all b i = 1 and D i are smooth divisors and they intersect transversally.(b) All F i j on D i in sequence (2.

Remark 4. 2 .
Let C be an integral curve andZ = {z 1 , • • • , z n } ⊂ C a finite subset.Then mult z i C ≥ 1 for each 1 ≤ i ≤ n.Set ν : C −→ Cto be the normalization.For a vector bundle E over C, consider the following fiber product diagram Bl η −1 (Z) P(E) = P(ν * E) is the projective bundle over C associated to E and η is the natural map.Let ξ and ξ ′ denote the numerical classes of O P(ν * E) (1) and O P(E) (1) respectively.Then ν * (ξ) = ξ ′ .

Example 4. 7 .
Let π : P C (W ) −→ C be a ruled surface over an elliptic curve C defined by the normalized rank 2 bundle W which sits in the following exact sequence:0 −→ O C −→ W −→ O C (p) −→ 0 for some point p ∈ C. Therefore the associated invariant e = − deg(W ) is −1 (see[Har3,  Proposition 2.8]).We consider the simple normal crossing divisor σ + f , where σ denotes the normalized section of π such that O P(W ) (σ) ≃ O P(W ) (1) and f denotes a fiber of the map π.
Let ρ : X := P(W ) −→ Z be the associated projective bundle on Z.Take D ⊂ Z to be a normal crossing divisor on Z, and let F * be a semistable parabolic bundle of rank r on Z with parabolic divisor D. Then ρ * F * is a parabolic semistable bundle on X with parabolic divisorD ′ = ρ * (D).Let D ′ = n i=1 D ′ i be the decomposition of D ′ into irreducible components.Let L * be a parabolic line bundle with parabolic divisor D ′ on X.So L * is given by a pair L, {α 1 , • • • , α n } , where L is a line bundle on X and 0 ≤ α i < 1 corresponds to the divisorD ′ i .Assume that α i ∈ Q for all i.Then E * = ρ * (F * ) ⊗ L * is parabolic semistable with △ par (E * ) = 0. Note that c 1 (L * ) := c 1 (L) + n i=1 α i [D i ].
filtration {E t } t∈R in (2.3) defines a parabolic structure on E. It is straightforward to check that all parabolic structures on E, with D as the parabolic divisor, arise this way.Let E * and W * be two parabolic sheaves on X, with D as the parabolic divisor, such that E 0 X\D and W 0 X\D are locally free.For any c ∈ R, define M c to be the subsheaf of φ * φ * (E ⊗ W ) generated by all E s ⊗ W t with s + t ≥ c, where φ is the map in (2.4).The parabolic sheaf given by the filtration M c c∈R is called the parabolic tensor product of E * and W * , and it is denoted by E * ⊗ W * .The parabolic m-fold symmetric product S m (E * ) is the invariant subsheaf of the m-fold parabolic tensor product of E * for the natural action of the permutation group for the factors of the tensor product.The underlying sheaf of the parabolic sheaf S m (E * ) will be denoted by S m (E * ) 0 .is called parabolic ample if for any coherent sheaf F on X there is an integer m 0 such that for any m ≥ m 0 , the tensor product F ⊗ S m (E * ) 0 is globally generated. *