Alexander Schmitt

Stability Parameters

STABILITY PARAMETERS FOR QUIVER SHEAVES

Keywords: 1991 Mathematics Subject Classification. Subject Classification: 14D20 (14F05, 16G20) Boundedness, Harder-Narasimhan filtration, instability flag, moduli, quiver, slope semistability

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Introduction

Hitchin used the technique of dimensional reduction of self-duality equations in four dimensions to derive interesting equations for a vector bundle over a compact riemann surface endowed with a holomorphic structure and an endomorphism ( [START_REF] Hitchin | The self-duality equations on a Riemann surface[END_REF], [START_REF] Hitchin | Gauge theory on Riemann surfaces[END_REF]). This led to the theory of higgs bundles on compact riemann surfaces. Likewise, García-Prada applied it to the hermite-einstein equations for a bundle on the product of a compact riemann surface and the projective line and found coupled vortex equations on the compact riemann surface whose solutions may be interpreted as holomorphic triples. In a purely algebraic context, homogeneous bundles on a homogeneous variety G/P , G a complex reductive algebraic group, P ⊂ G a parabolic subgroup, may be described in terms of representations of a(n) (infinite) quiver ( [START_REF] Bondal | Homogeneous bundles, in Helices and vector bundles[END_REF], [START_REF] Hille | Homogeneous vector bundles and Koszul algebras[END_REF], [START_REF] Álvarez-Cónsul | Dimensional reduction and quiver bundles[END_REF], [START_REF] Ottaviani | Quivers and the cohomology of homogeneous vector bundles[END_REF]). (This is a dimensional reduction from G/P to a point.) Álvarez-Cónsul and García-Prada generalized this to identify homogeneous vector bundles on the product of a compact kähler manifold and G/P with quiver bundles on the kähler manifold [START_REF] Álvarez-Cónsul | Dimensional reduction and quiver bundles[END_REF]. In that work, the concept of a quiver sheaf was developed. Furthermore, the papers [START_REF] Álvarez-Cónsul | Dimensional reduction and quiver bundles[END_REF] and [START_REF] Álvarez-Cónsul | Hitchin-Kobayashi correspondence, quivers, and vortices[END_REF] contain the parameter dependent notion of slope semistability for these objects. The homological algebra of quiver sheaves was investigated in [START_REF] Gothen | Homological algebra of twisted quiver bundles[END_REF]. Gieseker type notions of semistability for quiver sheaves were discussed in [START_REF] Schmitt | Moduli problems of sheaves associated with oriented trees[END_REF], [START_REF] Schmitt | Moduli for decorated tuples of sheaves and representation spaces for quivers[END_REF], [START_REF] Álvarez-Cónsul | Some results on the moduli spaces of quiver bundles[END_REF], [START_REF] Schmitt | A remark on semistability of quiver bundles[END_REF] and moduli spaces constructed. Quiver sheaves have discrete invariants of a topological flavor, namely the hilbert polynomials of the participating sheaves. For fixed values of these discrete invariants, we now have a family of moduli spaces, depending on a countable set of stability parameters. The present article is motivated by the study of the variation of the moduli spaces with the stability parameter. The basic property we will establish here is that there are only finitely many distinct semistability concepts and, so, only finitely many distinct moduli spaces. This continues our work in [START_REF] Álvarez-Cónsul | On the geometry of moduli spaces of holomorphic chains over compact Riemann surfaces[END_REF], [START_REF] Laudin | Recent results on quiver sheaves[END_REF], [START_REF] Schmitt | Global boundedness for semistable decorated principal bundles with special regard to quiver sheaves[END_REF]. In [START_REF] Andreas | Vector bundles in algebraic geometry and mathematical physics[END_REF], we discussed the chamber structure on the set of stability parameters which explains how moduli spaces associated with different stability parameters interact. The result of the present paper show that there are only finitely many chambers. In this way, we get a basic qualitative picture of the family of moduli spaces.

We will use the set-up of [START_REF] Andreas | Vector bundles in algebraic geometry and mathematical physics[END_REF]. Introductions to quiver sheaves are also contained in the references mentioned in the previous paragraph. We will work over a fixed polarized complex projective manifold (X, O X (1)). Fix a quiver Q = (V, A, t, h) and a tuple M = (M a , a ∈ A) of locally free O X -modules of finite rank. An M -twisted Q-sheaf is a tuple (E v , v ∈ V, ϕ a , a ∈ A) in which E v is a torsion free coherent O X -module, v ∈ V , and ϕ a : M a ⊗ E t(a) -→ E h(a) is a (twisted) homomorphism, a ∈ A. We will refer to the tuple (rk(E v ), v ∈ V, deg(E v ), v ∈ V ) as the type of (E v , v ∈ V, ϕ a , a ∈ A). The notion of slope semistability for M -twisted Q-sheaves depends on a tuple κ ∈ (R >0 ) ×#V and a tuple χ ∈ R #V . For such parameters κ = (κ v , v ∈ V ), χ = (χ v , v ∈ V ) and a tuple (F v , v ∈ V ) of coherent O X -modules, we define

• the κ-rank as rk κ (F v , v ∈ V ) := � v∈V κ v • rk(F v ),
• the (κ, χ)-degree as deg κ,χ

(F v , v ∈ V ) := � v∈V (κ v • deg(F v ) + χ v • rk(F v )),
• and, if the κ-rank is positive, the (κ, χ)-slope as

µ κ,χ (F v , v ∈ V ) := deg κ,χ (F v , v ∈ V ) rk κ (F v , v ∈ V ) . An M -twisted Q-sheaf (E v , v ∈ V, ϕ a , a ∈ A) is (κ, χ)-slope semistable, if the in- equality µ κ,χ (F v , v ∈ V ) ≤ µ κ,χ (E v , v ∈ V )
is satisfied for every non-trivial proper

Q-subsheaf (F v , v ∈ V ) of (E v , v ∈ V, ϕ a , a ∈ A).
If one fixes the type (n, d) and the stability parameter (κ, χ), then it is easy to show that there exists a constant C with

µ max (E v0 ) ≤ C,
for every vertex v 0 ∈ V and every (κ, χ)-slope semistable Q-sheaf (E v , v ∈ V, ϕ a , a ∈ A) of type (n, d) ( [START_REF] Schmitt | Geometric invariant theory and decorated principal bundles[END_REF], Exercise 2.5.4.1, [START_REF] Schmitt | A remark on semistability of quiver bundles[END_REF], Proposition 2.1). In [START_REF] Schmitt | Global boundedness for semistable decorated principal bundles with special regard to quiver sheaves[END_REF], we proved that the result still holds, if one fixes just (n, d) and κ and lets χ vary freely. In this paper, we will complete the picture and show:

Main Theorem. Let (X, O X (1)) be a polarized complex projective manifold, Q = (V, A, t, h) a quiver, M = (M a , a ∈ A) a tuple of twisting vector bundles, and (n, d) a type. Then, there exists a constant C, such that, for every stability parameter

(κ, χ) ∈ (R >0 ) ×#V × R #V , every (κ, χ)-slope semistable Q-sheaf (E v , v ∈ V, ϕ a , a ∈ A) of type (n, d)
, and every vertex v 0 ∈ V , the inequality

µ max (E v0 ) ≤ C is satisfied.
This result is crucial for exploring moduli spaces. First of all, it immediately implies:

Corollary. Let (X, O X (1)) be a polarized complex projective manifold, Q = (V, A, t, h) a quiver, M = (M a , a ∈ A) a tuple of twisting vector bundles, and P = (P v , v ∈ V ) a tuple of Hilbert polynomials. Then, the family of torsion free coherent O X -modules F for which there exist a stability parameter

(κ, χ), a (κ, χ)-slope semistable M -twisted Q-sheaf (E v , v ∈ V, ϕ a , a ∈ A) with P (E v ) = P v , v ∈ V , and a vertex v 0 ∈ V with F ∼ = E v0 is bounded.
To study slope semistability for quiver sheaves further, let us point out that several normalizations are possible. Obviously, for (κ, χ) ∈ (R >0 ) ×#V × R #V and λ ∈ R >0 , the notions of (κ, χ)-slope semistability and of (λ • κ, λ • χ)-slope semistability are equivalent. This means that we may require [START_REF] Álvarez-Cónsul | Some results on the moduli spaces of quiver bundles[END_REF] κ

∈ Ξ := � ν ∈ (R >0 ) ×#V � � kνk = 1 � .
Here, k • k stands for the maximum norm on R #V . Next, one may also check that one may choose

(κ, χ) ∈ Π := � (ν, ψ) ∈ Ξ × R #V � � � � v∈V (ν v • d v + ψ v • r v ) = 0 � .
Using this normalization, we defined in [START_REF] Andreas | Vector bundles in algebraic geometry and mathematical physics[END_REF], Section 2.5, a locally finite subdivision of Ξ × R #V into locally closed chambers, such that the notion of (κ, χ)-slope semistability remains constant within each chamber. If we combine [START_REF] Andreas | Vector bundles in algebraic geometry and mathematical physics[END_REF], Proposition 2.6.1, with the main result of the present article, we obtain:

Theorem. In the situation of the corollary, there is a finite set { (κ 1 , χ 1 ), ...,

(κ s , χ s ) } ⊂ (R >0 ) ×#V × R #V of stability parameters, such that, for any (κ, χ) ∈ (R >0 ) ×#V × R #V
, there is an index j ∈ { 1, ..., s }, such that the notions of (κ, χ)slope (semi)stability and (κ j , χ j )-slope (semi)stability for

M -twisted Q-sheaves (E v , v ∈ V, ϕ a , a ∈ A) with P (E v ) = P v , v ∈ V , are equivalent.
As usual, the chamber decomposition gives an indication how the moduli associated with different stability parameters interact (see, e.g., [START_REF] Álvarez-Cónsul | On the geometry of moduli spaces of holomorphic chains over compact Riemann surfaces[END_REF], Section 6, [START_REF] Laudin | Recent results on quiver sheaves[END_REF], Section 3.5).

Let us comment a bit on the techniques in this paper. Quiver sheaves are examples of decorated principal bundles in the sense of [START_REF] Schmitt | Geometric invariant theory and decorated principal bundles[END_REF]. Recall that a rational principal bundle on X is a pair (U, P) in which U ⊂ X is a big open subset1 and P is a principal G-bundle on U . For example, if E is a torsion free coherent O Xmodule of rank n ≥ 1, then the locus U where E is locally free is a big open subset, and the frame bundle of E |U is a principal GL n (C)-bundle, so that E gives rise to a rational principal GL n (C)-bundle. We fix a representation σ : G -→ GL(H) and a line bundle L on X. Then, an L -twisted affine σ-bump is a pair ((U, P), ϕ) which consists of a rational principal G-bundle (U, P) on X and a homomorphism ϕ : P σ -→ L |U . Here, P σ is the vector bundle on U with typical fiber H that is associated with P and the G-action on H induced by σ. Let us remind you of the significance of the three different types of stability parameters that we have used above. The first one is the choice of a polarization O X (1) on X. This is irrelevant, if X is a curve, and is quite tricky to understand, if dim(X) ≥ 3, even if one deals "just" with vector bundles or torsion free coherent sheaves ( [START_REF] Zh | Equivalence classes of polarizations and moduli spaces of sheaves[END_REF], [START_REF] Schmitt | Walls for Gieseker semistability and the Mumford-Thaddeus principle for moduli spaces of sheaves over higher dimensional bases[END_REF], [START_REF] Greb | Variation of Gieseker-Maruyama moduli spaces via quiver GIT[END_REF]). We will not talk about it in this paper. The second parameter is the choice of a character χ of G. This appeared already in the first examples of decorated vector bundles [START_REF] Bradlow | Special metrics and stability for holomorphic bundles with global sections[END_REF] and was treated in a general manner in [START_REF] Schmitt | Global boundedness for semistable decorated principal bundles with special regard to quiver sheaves[END_REF]. The last parameter is the choice of a faithful representation κ : G -→ GL(K). More generally, one may fix a maximal torus on T ⊂ G and a Weyl invariant pairing on X ⋆ (T ) ⊗ Z C (compare [START_REF] Lübke | The universal Kobayashi-Hitchin correspondence on Hermitian manifolds[END_REF]). To my knowledge, this parameter has not been systematically studied in the algebro geometric context. In this paper, we will make a first general contribution. So far, two methods have been used in study boundedness questions for (decorated) principal bundles. The first one, appearing in [START_REF] Schmitt | Global boundedness for decorated sheaves[END_REF], [START_REF] Schmitt | Global boundedness for semistable decorated principal bundles with special regard to quiver sheaves[END_REF], is based on the formalism of the instability flag, in particular, on its application to the study of semistable rational principal bundles by Ramanan and Ramanathan [START_REF] Ramanan | Some remarks on the instability flag[END_REF]. The second method was introduced in [START_REF] Gómez | Moduli spaces for principal bundles in arbitrary characteristic[END_REF] and works for direct sums of tensor powers of the standard representation of GL n (C). It is technically much easier. Note that the second method cannot be directly applied in the present sitation, because the estimates one gets with this approach involve terms of the form χ v /κ v (see [START_REF] Andreas | Vector bundles in algebraic geometry and mathematical physics[END_REF], Proposition 2.4.2, or [START_REF] Schmitt | Moduli for decorated tuples of sheaves and representation spaces for quivers[END_REF], Remark 3.3.2), for some vertices v of V , and these terms might get arbitrarily large, if κ approaches the boundary of Ξ. 2 An important observation we make here is that this approach still works well, if χ = 0 (see the proof of Theorem 4.2). In order to control χ, we recur again to the mechanism of the instability flag. The latter depends on the choice of a Weyl invariant norm on the space of real characters of a maximal torus of G. In our set-up, G is a direct product of general linear groups, and the norm corresponds to the stability parameter κ. We need to control the instability flag when κ approaches the boundary of Ξ. This requires some refined estimates. Maybe this is the first time that the dependence of the instability on the chosen Weyl invariant norm has an impact.

Conventions. We will use freely the notation and the results from the papers [START_REF] Andreas | Vector bundles in algebraic geometry and mathematical physics[END_REF], [START_REF] Schmitt | Global boundedness for semistable decorated principal bundles with special regard to quiver sheaves[END_REF]. In addition, we will adopt the following terminology: Given a tuple k = (k 1 , ..., k w ) of integers, we set |k| := k 1 + • • • + k w . For a natural number n ∈ N, the symbol [n] is a short-hand for the set { 1, ..., n }. We will assume that the vertex set V of the quiver Q is of the form [w] for some natural number w ≥ 1. The letter K denotes the function field C(X) of X. For w ∈ N, the symbol R w + stands for (R >0 ) ×w , the space of w-tuples of positive real numbers.

If we have fixed Q, M , L , as well as the type (n, d), and if we speak of "constants", "bounds" or similar concepts, it is understood that they do only depend on these background data.

{ e 1 , ..., e n } of X ⋆ (D), we introduce the pairng

(•, •) ⋆ : X ⋆ (D) × X ⋆ (D) -→ Z � n � i=1 χ i • e i , n � i=1 ψ i • e i � 7 -→ n � i=1 χ i • ψ i .
There is a similar pairing (•, •) : X(D) × X(D) -→ Z.

Next, we introduce the real vector spaces

X ⋆,R (D) := X ⋆ (D) ⊗ Z R and X R (D) := X(D) ⊗ Z R. The pairings (•, •) ⋆,R : X ⋆,R (D) × X ⋆,R (D) -→ R, h•, •i R : X R (D) × X ⋆,R (D) -→ R, and (•, •) R : X R (D)×X R (D) -→ R
∨ := n � i=1 χ i • e i ∈ X R (D).
In terms of the above pairings, l ∨ is characterized by the property

∀l ′ ∈ X ⋆,R (D) : (l, l ′ ) ⋆,R = hl ∨ , l ′ i R .
Now, let w ≥ 1 be a natural number, n = (n 1 , ..., n w ) a tuple of positive natural numbers, and κ = (κ 1 , ..., κ w ) a tuple of non-negative real numbers. We set

GL n (C) := w ą i=1 GL ni (C), let D i ⊂ GL ni (C) be the maximal torus consisting of diagonal matrices and D = w Ś i=1 D i ⊂ GL n (C). Then, X ⋆,R (D) = w � i=1 X ⋆,R (D i ) and X R (D) = w � i=1 X R (D i ).
So, we write a real cocharacter l ∈ X ⋆,R (D) as a tuple l = (l 1 , ..., l w ) with l i ∈ X ⋆,R (D i ), i ∈ [w]. We use an analogous notation for real characters of D.

Next, introduce the pairing

(•, •) ⋆,κ : X ⋆,R (D) × X ⋆,R (D) -→ R � (l 1 , ..., l w ), (l ′ 1 , ..., l ′ w ) � 7 -→ w � i=1 κ i • (l i , l ′ i ) ⋆,R ,
and, similarly,

(•, •) κ : X R (D) × X R (D) -→ R.
Note that these pairings become degenerate, if not all the entries of κ are positive. The induced (semi-)norms will be denoted by

k • k ⋆,κ : X ⋆,R (D) -→ R, and k • k κ : X R (D) -→ R.
The natural pairing between characters and cocharacters is

h•, •i R : X R (D) × X ⋆,R (D) -→ Z � (x 1 , ..., x w ), (l 1 , ..., l w ) � 7 -→ w � i=1 hx i , l i i R .
For a real cocharacter l = (l 1 , ..., l w ) ∈ X ⋆,R (D), we define the dual real character

l ∨ κ := (κ 1 • l ∨ 1 , ..., κ w • l ∨ w ) ∈ X R (D). The formula ∀l ′ ∈ X ⋆,R (D) : (l, l ′ ) ⋆,κ = hl ∨ κ , l ′ i R still holds true.
Remark 1.1. Suppose that κ = (κ 1 , ..., κ w ) is a tuple of positive integers. Then, we have the embedding

ι κ : GL n (C) -→ GL N (C), N := κ 1 • n 1 + • • • + κ w • n w ,
that maps a tuple (m 1 , ..., m w ) of matrices to the block diagonal (N × N )-matrix in which the block m 1 is first repeated κ 1 times, then the block m 2 κ 2 times, and so on. In this case, the pairing (•, •) ⋆,κ and the norm k • k ⋆,κ on X ⋆,R (D) are simply those induced by the corresponding pairing and norm on X ⋆,R (T ), T ⊂ GL N (C) the maximal torus of diagonal matrices that were described at the beginning.

A refined analysis of the instability flag

In this section, we will adopt the notation of Section 2 of [START_REF] Schmitt | Global boundedness for semistable decorated principal bundles with special regard to quiver sheaves[END_REF]. Abbreviate G := GL n (C). Fix a Borel subgroup D ⊂ B ⊂ G. We start with a finite dimensional complex vector space H and a representation σ : G -→ GL(H). Then, we form

R := H ⊕ C and ρ := σ ⊕ �: G -→ GL(R).
We need to investigate the instability one parameter subgroups of points of the shape r

= (h, 1), h ∈ H. Let χ ∈ X R (G) := X(G) ⊗ Z R be a real character of G. The point r is χ-(semi)stable, if ∀g ∈ G ∀l ∈ C (B, D) : µ St D (σ(g)(h)) (l) ≤ 0 =⇒ hχ, li R (≥)0. For the following, fix a norm k • k on X R (G). Note that, for η ∈ R >0 , the condition of χ-(semi)stability is equivalent to the one of (η • χ)-(semi)stability. Proposition 2.1. There is a constant K > 0, such that, for a character χ ∈ X R (G) with kχk < 1 K ,
the following conditions are equivalent:

• The point r is χ-(semi)stable. • The point [r] = [h, 1] ∈ P (R) is GIT-(semi)
stable with respect to the natural linearization of the G-action in O P (R) (1) twisted by the character χ.

This is Corollary 2.11 in [START_REF] Schmitt | Global boundedness for semistable decorated principal bundles with special regard to quiver sheaves[END_REF]. For subsequent arguments, we will have to review the proof of this result.

Proof. First note that, for any character χ ∈ X R (G), the second condition implies the first one. Let ( 2)

C (B, D) = h � i=1 B i
be the decomposition of the Weyl chamber into cones described in [START_REF] Schmitt | Global boundedness for semistable decorated principal bundles with special regard to quiver sheaves[END_REF], p. 457, and

T = { λ 1 , ..., λ m }
the set of minimal integral generators of edges of these cones. Introduce

L i := max � -hψ, λ i i R � � ψ ∈ X R (G) ∧ kψk = 1 � , i ∈ [m], L := max{ L 1 , ..., L m }.
Next, let S be the set of states of the representation σ. Then, we pick a positive natural number M with the property

∀ω ∈ S , i ∈ [m] : hω, λ i i 6 = 0 =⇒ � � hω, λ i i � � > 1 M .
The proposition is true for

K := M • L. �
We need some more data. Let k • k be a norm on X ⋆,R (D) and set

Υ := � l ∈ X ⋆,R (D) � � klk = 1 � .
Furthermore, we fix a finite subset Γ ⊂ G with the property

(3) � St D (σ(γ)(h)) | γ ∈ Γ � = � St D (σ(g)(h)) | g ∈ G � .
Pick a stability parameter κ ∈ Ξ,3 let χ ∈ X R (D), and assume that the point r = (h, 1) is χ-unstable. We call a point (γ 0 , l 0 ) ∈ Γ × Υ at which the function

ν χ,κ : Γ × Υ -→ R (γ, l) 7 -→ µ(ρ(γ)(r), l) + hχ, li R klk ⋆,κ
attains its minimum an instability point of r and l r := γ -1 0 • l 0 • γ 0 an instability one parameter subgroup of r. In [START_REF] Schmitt | Global boundedness for decorated sheaves[END_REF], Theorem 2.10, and [START_REF] Schmitt | Global boundedness for semistable decorated principal bundles with special regard to quiver sheaves[END_REF], Theorem 3.9, the following result was obtained: Proposition 2.2. There is a constant K κ , such that, for a stability parameter χ ∈ X R (G) with kχk < 1/K κ and a χ-unstable point r = (h, 1) ∈ R, an instability point (γ 0 , l 0 ) and the instability one parameter subgroup l r satisfy µ(r, l r ) = µ(ρ(γ 0 )(r), l 0 ) = 0.

As indicated by the notation, the constant K κ depends on κ. Unfortunately, there is no uniform version of this result. For this reason, we need more precise information about K κ . First, we note that we can determine (γ 0 , l 0 ) without any restriction on χ. To this end, we look at the closed subset

(Γ × Υ ) r := � (γ, l) ∈ Γ × Υ | µ(ρ(γ)(r), l) = 0 � of Γ × Υ . Let (γ 0 , l 0 ) ∈ (Γ × Υ
) r be a point at which the function ν κ,χ takes its minimal value. By Proposition 2.2, (γ 0 , l 0 ) is an instability point of r with respect to (1/N ) • χ, for all N ≫ 0. In particular, γ -1 0 • l 0 • γ 0 is uniquely defined, and χ and η • χ yield the same point, η ∈ R >0 .

Next, we remind the reader of the salient feature of an instability point (γ 0 , l 0 ). First, l 0 defines a filtration

{0} = R 0 ( R 1 ( • • • ( R s-1 ( R s = R. Let Q G (l 0 ) ⊂ G be the closed subgroup that stabilizes this filtration and L G (l 0 ) ⊂ Q G (l 0 ) the centralizer of l 0 . It is a Levi subgroup of Q G (l 0 ). Remark 2.3. Note that L G (l 0 ) ∼ = s ą j=1 GL(R j /R j-1 ).
In particular, there are only finitely many possibilities for L G (l 0 ) as the parameters vary, call them H 1 , ..., H t .

Let j(r) be maximal among the indices j ∈ [s] for which the image of ρ(γ 0 )(r) in R j /R j-1 is non-zero and r the image of ρ(γ 0 )(r) in R j(r) /R j(r)-1 . Remark 2.4. i) The vector space R j(r) /R j(r)-1 is in a natural way an L G (l 0 )module. Again, there are only finitely many options for this module. In the notation of Remark 2.3, these modules give rise to finitely many representations denoted by

ρ i j : H i -→ GL(S i j ), j ∈ [u i ], i ∈ [t]. ii) Since we have µ(ρ(γ 0 )(r), l 0 ) = 0, the representation R j(r) /R j(r)-1 contains the trivial representation � as a direct summand. Let � χ := l ∨ 0,κ ∈ X R (D) be the dual real character of l 0 (see Page 5), i.e., � χ satisfies (4) ∀l ∈ X ⋆,R (D) : h� χ, li R = (l 0 , l) ⋆,κ .
Finally, we set

(5) χ r := h� χ, l 0 i R • � χ.
The following is a crucial result of Ramanan and Ramanathan ( [START_REF] Ramanan | Some remarks on the instability flag[END_REF], Proposition 1.12):

Proposition 2.5. Assume χ ∈ X R (D) is a character with kχk < K κ . The point [r] ∈ P (R j(r) /R j(r)-1
) is GIT-semistable with respect to the natural linearization of the L G (l 0 )-action in O P (R j(r) /R j(r)-1 ) (1) twisted by the character

χ + χ r kl 0 k 2 ⋆,κ . Remark 2.6. If η ∈ R >0 is a positive constant and χ η r is the character from (5) associated with η • χ, then η • χ + χ η r kl 0 k 2 ⋆,κ = η • � χ + χ r kl 0 k 2 ⋆,κ � .
As we explained before, we need not assume kχk < 1/K κ for defining (γ 0 , l 0 ) and r. Moreover, if χ ∈ X R (G) is any character and (γ 0 , l 0 ) ∈ (Γ × Υ ) r is a point where ν κ,χ takes on its minimal value, then the associated point r ∈ R j(r) /R j(r)-1 verifies the condition of semistability with respect to the character

χ + χ r kl 0 k 2 ⋆,κ , for all points (h, l) ∈ H i × X ⋆,R (D) with µ(ρ i j (h)(r), l) = 0. Here, i ∈ [t], j ∈ [u i ] are the indices with H i = L G (l) and ρ i j "="R j(r) /R j(r)-1 . Remark 2.4, ii), shows that µ(ρ i j (h)(r), l) ≥ 0 holds for any (h, l) ∈ H i × X ⋆,R (D). It remains to look at those (h, l) ∈ H i × X ⋆,R (D) with µ(ρ i j (h)(r), l) > 0.
As usual, one may restrict to a finite number of such (h, l)'s:

Observe that D is a maximal torus of H i , i ∈ [t]. We fix a Borel subgroup D ⊂ B i ⊂ H i , i ∈ [t]
. Then, we may find a decomposition

C (B i , D) = mi � k=1 B i k as in (2), working for H i and ρ i 1 , ..., ρ i ui , i ∈ [t]
. We add the one parameter subgroups of D which occur as minimal integral generators of one of the above cones to T and obtain the larger set

T ⊂ T ′ = { λ 1 , ..., λ v }.
We have to find out how negative hχ r ,

λ i i R /kl 0 k ⋆,κ can get, i ∈ [v].
To this end, note that, for i ∈ [v], (4) and the Cauchy-Schwarz inequality give

hχ r , λ i i R = hχ, l 0 i R • (l 0 , λ i ) ⋆,κ ≥ hχ, l 0 i R • kl 0 k ⋆,κ • kλ i k ⋆,κ .
We set4

E i := sup � kλ i k ⋆,ν | ν ∈ Ξ � , E := max{ E 1 , ..., E v }, and (6)
F i := min � hψ, λ i i R | ψ ∈ X R (G) : kψk = 1 � , F := -min{ F 1 , ..., F v }.
Altogether, we find hχ,

λ i i R + hχ r , λ i i R kl 0 k 2 ⋆,κ ≥ -kχk • � F - hχ, l 0 i R kχk • kl 0 k κ • E � , i ∈ [v].
For each i ∈ [t] and each j ∈ [u i ], the set of states of ρ i j is contained in S ∪ {0}. Recall that S stands for the set of states of σ. As before, we let M ′ be a positive natural number with the property

∀ω ∈ S , i ∈ [v] : hω, λ i i 6 = 0 =⇒ � � hω, λ i i � � > 1 M .
Invoking Remark 2.6, we obtain the following variant of Proposition 2.5.

Proposition 2.7. Assume χ ∈ X R (D) is a character with kχk = 1, κ ∈ Ξ, and r = (h, 1) ∈ R is a χ-unstable point. Let (γ 0 , l 0 ) ∈ (Γ × Υ ) r be a point at which v χ,κ|(Γ ×Υ )r attains its minimal value and

0 < η < 1 M • � F -E • hχ, l 0 i R kl 0 k ⋆,κ � a positive real number. Then, the point [r] ∈ P (R j(r) /R j(r)-1 ) is GIT-semistable with respect to the natural linearization of the L G (l 0 )-action in O P (R j(r) /R j(r)-1 ) (1)
twisted by the character

η • � χ + χ r kl 0 k 2 ⋆,κ � .
Remark 2.8. Let Λ ⊂ D be the subgroup of scalar matrices and assume that Λ acts trivially on H. This is, for example, the case, if H is the space Rep(Q, n) := � a∈A Hom(C n t(a) , C n h(a) ) of representations of a dimension vector n of a quiver Q.

Next, let κ = (κ 1 , ..., κ w ) be a tuple of positive integers and ι κ : GL n (C) -→ GL N (C) the embedding described in Remark 1.1. Set

D ′ := D ∩ SL N (C). Then, X ⋆,R (D) = X ⋆,R (D ′ ) ⊕ X ⋆,R (Λ) ∼ = X ⋆,R (D ′ ) ⊕ R. Note that X ⋆,R (D ′ ) and X ⋆,R (Λ) are orthogonal with respect to the pairing (•, •) ⋆,κ introduced in Section 1.
Next, let χ = (χ 1 , ..., χ w ) be a tuple of rational numbers subject to the condition

(7) w � i=1 χ i • n i = 0.
This means exactly that the rational character χ of GL n (C) associated with χ is trivial on Λ. Now, let r = (h, 1) ∈ R be a χ-unstable point and l r ∈ X ⋆,Q (D) an instability one parameter subgroup with respect to the character χ. We suppose µ(r, l r ) = 0. We decompose

l r = l ′ + l ′′ with l ′ ∈ X ⋆,Q (D ′ ) and l ′′ ∈ X ⋆,Q (Λ). Then, hχ, l r i R = hχ, l ′ i R and kl r k 2 ⋆,κ = kl ′ k 2 ⋆,κ + kl ′′ k 2 ⋆,κ
. This shows that l ′′ = 0, i.e., l r ∈ X ⋆,R (D ′ ).

Boundedness of some of the stability parameters

As in [START_REF] Schmitt | Global boundedness for semistable decorated principal bundles with special regard to quiver sheaves[END_REF], we will use the ideas of Ramanan and Ramanathan [START_REF] Ramanan | Some remarks on the instability flag[END_REF] in order to obtain bounds on the stability parameters. The theory of semistable decorated principal bundles on curves has been described in [START_REF] Schmitt | Geometric invariant theory and decorated principal bundles[END_REF]. The extension of the basic notions to rational principal bundles on higher dimensional projective manifolds has been outlined in [START_REF] Schmitt | Geometric invariant theory and decorated principal bundles[END_REF], Section 2.9.2, and [START_REF] Schmitt | Global boundedness for semistable decorated principal bundles with special regard to quiver sheaves[END_REF], Section 1. We continue to look at the structure group G = GL n (C) and a representation σ : G -→ GL(H) which is homogeneous of degree zero in the sense of [START_REF] Schmitt | Moduli for decorated tuples of sheaves and representation spaces for quivers[END_REF], Section 3.1. As before, we fix m > 0 and the line bundle O X (m). Recall that an O X (m)-twisted affine σ-bump is a pair ((U, P), ϕ) which consists of a rational principal G-bundle (U, P) on X and a homomorphism ϕ

: P σ -→ O X (m) |U . Next, fix an injective homomorphism ε : O X -→ O X (m).
With an O X (m)-twisted affine σ-bump ((U, P), ϕ), we associate the O X (m)-twisted affine ρ-bump ((U, P), ψ) with ψ := ϕ ⊕ ε |U :

P ρ = P σ ⊕ O U -→ O X (m) |U .
In the introduction, we spoke about the normalization

w � i=1 (κ i • d i + χ i • n i ) = 0.
Here, we will need another one. Set [START_REF] Bradlow | Special metrics and stability for holomorphic bundles with global sections[END_REF] N :=

� ψ = (ψ 1 , ..., ψ w ) ∈ R w � � w � i=1 ψ i • n i = 0 � .
As recalled in the introduction, we may restrict to stability parameters (κ, χ) with kκk = 1 and χ ∈ N . The same holds in general ([25], Remark 2.5.3.5, ii). Note that an element χ ∈ N may be viewed as a real character of G which is trivial on G m embedded diagonally.

Let us briefly review the definition of (semi)stability. Fix κ ∈ R w + and χ ∈ N . Write δ i : G -→ G m for the character

(g i , i ∈ [w]) 7 -→ det(g i ), i ∈ [w]. We set X ⋆,R (D) κ-SL := � (l 1 , ..., l w ) ∈ X ⋆,R (D) � � w � i=1 κ i • hδ i , l i i R = 0
� .

An element l ∈ X ⋆,R (D) κ-SL defines a weighted flag (W • , γ) (compare [START_REF] Schmitt | Moduli for decorated tuples of sheaves and representation spaces for quivers[END_REF], Section 3.2) in the [w]-split vector space (C ni , i ∈ [w]). Note that the entries of the tuple γ may be real numbers in our case. The flag W • defines a parabolic subgroup of G that will be denoted by Q G (l). Now, let (U, P) be a rational principal G-bundle on X and (E i , i ∈ [w]) the corresponding [w]-split vector bundle on U . Suppose l ∈ X ⋆,R (D) κ-SL . A reduction of (U, P) to l is a section

β : U ′ -→ P |U ′ /Q G (l)
over a big open subset U ′ ⊂ U . It corresponds to a filtration

E • : 0 ( (E 1 i , i ∈ [w]) ( • • • ( (E s i , i ∈ [w]) ( (E i|U ′ , i ∈ [w]) of the [w]-split vector bundle (E i|U ′ , i ∈ [w]) on U ′ where the rank of E j i agrees with the dimension of W j i from the filtration W • , i ∈ [w], j ∈ [s]
. Let γ := (γ 1 , ..., γ s+1 ) be the tuple of real numbers from the weighted flag (W • , γ) associated with l and set

α i := γ i+1 -γ i w � i=1 κ i • n i , i ∈ [s], α := (α 1 , ..., α s ).
Next, we define

L κ,χ (E • , α) := s � j=1 α j • � deg κ,χ � E i , i ∈ [w] � •rk κ � E j i , i ∈ [w] � -deg κ,χ � E j i , i ∈ [w] � •rk κ � E i , i ∈ [w] � � .
Remark 3.1. Suppose that κ is a tuple of positive integers. i) As described in Remark 1.1, there is a corresponding faithful representation ι κ : G -→ GL N (C). Given a genuine one parameter subgroup λ ∈ X ⋆ (D)∩SL N (C), the pairing (•, •) ⋆ may be used to define a character χ λ on Q G (λ). A reduction β of (U, P) to λ gives a principal Q G (λ)-bundle on U ′ . This bundle and χ λ define a line bundle L β on U ′ . The number L κ,0 (E • , α) does compute the degree of that line bundle (see [START_REF] Schmitt | Geometric invariant theory and decorated principal bundles[END_REF], Exercise 2.4.9.2), 5 and there is the following identity

L κ,χ (E • , α) = L κ,0 (E • , α) + hχ, λi = deg(L β ) + hχ, λi.
For this, we refer to [START_REF] Schmitt | Geometric invariant theory and decorated principal bundles[END_REF], Exercise 2.5.2.4 and Remark 2.5.3.5, i).

ii) As in Remark 2.8, the decomposition We continue with the above notation. Assume that ϕ : P σ -→ O X (m) |U is an O X (m) |U -twisted homomorphism. The definition of µ σ (β, ϕ) given in [START_REF] Schmitt | Global boundedness for semistable decorated principal bundles with special regard to quiver sheaves[END_REF], p. 449, makes sense in the more general set-up we are currently discussing.

X ⋆,R (D) = X κ-SL ⋆,R (D) ⊕ X ⋆,R (Λ) is orthogonal with respect to the pairing (•, •) ⋆,κ . Suppose l ∈ X ⋆,R (Λ). Then, Q G (l) = G and β := id U : U -→ U = P/G is a reduction with µ σ (β, ϕ) = 0.
An O X (m)-twisted affine σ-bump

((U, P), ϕ) is (κ, χ)-(semi)stable, if the in- equality L κ,χ (E • , α)(≥)0
holds for every λ ∈ X ⋆,R (D) κ-SL and every reduction β :

U ′ -→ P |U ′ /Q G (λ) with µ σ (β, ϕ) ≤ 0.
Remark 3.2. i) Let ((U, P), ϕ) be an O X (m)-twisted affine σ-bump and ((U, P), ψ) the associated O X (m)-twisted affine ρ-bump. Then, for λ and β as before,

µ ρ (β, ψ) = max � µ σ (β, ϕ), 0 � . This means that ((U, P), ϕ) is (κ, χ)-(semi)stable if and only if ((U, P), ψ) is. ii) Let κ ∈ R w + , χ ∈ N and η > 0.
It is obvious that the notions of (κ, χ)-(semi)stability and (η • κ, η • χ)-(semi)stability are equivalent. In particular, if κ and χ are rational, they may be replaced by integral parameters. Then, we are in the usual set-up.

iii) Assume that κ is integral and ι κ : G -→ GL N (C) is the corresponding faithful representation. The arguments of [START_REF] Schmitt | Geometric invariant theory and decorated principal bundles[END_REF], Example 1.5.1.18, show that the notion of (semi)stability has to be checked only for finitely many conjugacy classes of (integral) one parameter subgroups. In particular, the notion of (κ, χ)-(semi)stability has to be checked only for (integral) one parameter subgroups λ ∈ X ⋆ (D)∩SL N (C).

iv) Let Q be a quiver and H = Rep(Q, n) the space of representations of Q with dimension vector n. Then, the above concept of (semi)stability is the same as the one presented in the introduction. This results from [START_REF] Schmitt | Geometric invariant theory and decorated principal bundles[END_REF], Proposition 1.5.1.22. In [START_REF] Andreas | Vector bundles in algebraic geometry and mathematical physics[END_REF], Remark 2.5.4, we checked that it is sufficient to look at rational stability parameters. A similar result holds for arbitrary representations, but the arguments are more tedious.

The idea is to apply the theory of the instability flag at the generic point of X. Let ((U, P), ϕ) be an O X (m)-twisted affine σ-bump. Since G is a special group, 7we may choose a trivialization

P |Spec(K) ∼ = G K := G × Spec(K) Spec(K). This induces a trivialization P σ|Spec(K) ∼ = H K := H ⊗ C K. The decoration ϕ yields a point ϕ K ∈ H K and ϕ ⊕ ε |U a point ψ K ∈ R K , R K := R ⊗ C K = H K ⊕ K.
For a tuple χ ∈ N , we may speak about χ-(semi)stability of ψ K with respect to the G K -action on R. We say that ((U, P), ϕ) is generically χ-(semi)stable, if ψ K is χ-(semi)stable in the sense of Section 2. If ((U, P), ϕ) is not generically χ-semistable, then it is said to be generically χ-unstable. We define ( 9)

Ω := � ψ ∈ N � � kψk = 1 � .
Suppose Ω ′ ⊂ Ω is a closed subset. We call ((U, P), ϕ) generically Ω ′ -unstable, if it is generically χ-unstable, for all χ ∈ Ω ′ . If ((U, P), ϕ) is generically Ω-unstable, we also say that it is generically totally unstable. For later purposes, we let

Ω ′ Q := Ω ′ ∩ (R >0 • Q w ) and Ξ Q := Ξ ∩ (R >0 • Q w ) consist of
those elements that generate rays containing rational points.

Last but not least, we need a topological invariant. We assign to a principal Gbundle P on X the tuple

(n i , i ∈ [w], d i , i ∈ [w]
) consisting of the ranks and degrees of the vector bundles in the corresponding [w]-split vector bundle (E i , i ∈ [w]). We define the function

d P : R w -→ R ψ = (ψ 1 , ..., ψ w ) 7 -→ w � i=1 ψ i • d i .
Theorem 3.3. Fix the type (n, d), and let Ω ′ ⊂ Ω be a closed subset. Then, there is a constant C 1 , depending only on (n, d), σ, and Ω ′ , such that, for rational stability

parameters κ ∈ Ξ Q , χ ∈ R w \ {0} with χ/kχk ∈ Ω ′ Q , the existence of a (κ, χ)-slope semistable O X (m)-twisted affine σ-bump in which (U, P) has type (n, d) and which is generically Ω ′ -unstable implies kχk ≤ C 1 .
Proof. To ease notation and to use the same notation as in Section 1 and 2, we write χ instead of χ. We use elements of the proof of Theorem 3.1 in [START_REF] Schmitt | Global boundedness for decorated sheaves[END_REF] and Theorem 4.2 in [START_REF] Schmitt | Global boundedness for semistable decorated principal bundles with special regard to quiver sheaves[END_REF]. Define χ := χ/kχk. Since the point ψ K is χ-unstable, it defines an instability point (γ 0 , l 0 ) ∈ ( � Γ K × Υ ) r . Here, K is the algebraic closure of K and

� Γ K ⊂ G K := G × Spec(C)
K is an appropriate finite subset (compare (3)).

The problem in the proof is that the decisive quantity hχ, l 0 i R /kl 0 k ⋆,κ could get arbitrarily negative when κ approaches the boundary of Ξ. So, we may assume without loss of generality8 that

- hχ, l 0 i R kl 0 k ⋆,κ > F,
F as defined in ( 6), and apply Proposition 2.7 with

η := -1 2 • M • E • kl 0 k ⋆,κ hχ, l 0 i R .
Now, we pick a natural number T > 0, such that

T • χ, T • κ, T • l 0 , N := T η are all integral.
Note. Let � χ r be defined with respect to T • l 0 and T • κ. Then,

h� χ r , T • l 0 i kT • l 0 k 2 ⋆,T •κ = T 4 • hχ r , l 0 i R T 3 • kl 0 k 2 ⋆,κ = T • hχ r , l 0 i R kl 0 k 2 ⋆,κ
.

We infer from Proposition 2.7 that [ψ K ] ∈ P (R K ), R K := (R j(r) /R j(r)-1 ) ⊗ C K, is semistable with respect to the linearization in O P (R K ) (N ) twisted by the character

T • � χ + χ r kl 0 k 2 ⋆,κ � .
As we pointed out after Proposition 2.2, l := γ -1 0 • l 0 • γ 0 is a genuine instability one parameter subgroup for ψ K . Thus, T • l ∈ G K . Note also that l 0 ∈ X ⋆,R (D ′ ) = X ⋆,R (D) κ-SL , by Remark 2.8. Using the trivialization of P |Spec(K) , we get a point

β K : Spec(K) -→ P |Spec(K) /Q G (T • l 0 ).
This can be extended to a reduction

β : U ′ -→ P |U ′ /Q G (T • l 0 )
over a big open subset U ′ ⊂ U ⊂ X. The argument of [START_REF] Schmitt | Global boundedness for decorated sheaves[END_REF], (3.21), and [START_REF] Schmitt | Global boundedness for semistable decorated principal bundles with special regard to quiver sheaves[END_REF], p. 479f, implies that

- 1 T • hχ, l 0 i R kl 0 k 2 ⋆,κ • deg(L β ) ≤ T • � d P (-χ) + 1 η • m • deg � O X (1) � � .
The continuous function ψ 7 -→ d P (-ψ) admits a maximum K 1 on the set Ω ′ , and

1 η • kl 0 k 2 ⋆,κ hχ, l 0 i R = -2 • M • E • kl 0 k ⋆,κ .
We find

deg(L β ) ≤ T 2 • K 2 • kl 0 k ⋆,κ , K 2 := - K 1 F -2 • M • E • m • deg � O X (1) � . Set K 3 := max � kλ i k ⋆,ν � � i ∈ [m], ν ∈ Ω � . Let R K := R ⊗ C K, G K := G ⊗ Spec(C)
Spec(K), and ρ K the representation of G K on R K . We view ψ K as an element of R K and set (10)

S := � St D � ρ K (� γ)(Ψ K ) � � � � γ ∈ � Γ K � .
For every S ∈ S, let I(S)

:= { i ∈ [m] | µ S (λ i ) = 0 }, and
I := � S∈S I(S).
Then,

Ω ′ -→ R ψ 7 -→ min � hψ, λ i i R � � i ∈ I �
is a continuous function, and the assumption implies that it has a negative maximum. Since there are only finitely many possibilities for S, we may find a constant K 4 > 0 which depends only on σ, such that this maximum is smaller than -K 4 .

For i ∈ I, we have hχ,

l 0 i R kl 0 k ⋆,κ ≤ hχ, λ i i R kλ i k ⋆,κ .
It follows readily that

(11) hχ, l 0 i R kl 0 k ⋆,κ ≤ - K 4 K 3 .
Recall (Remark 3.2, ii) that a ρ-bump which is (κ, χ)-slope semistable is also (T • κ, T • χ)-slope semistable. In particular, we find

0 ≤ deg(L β ) + hT • χ, T • l 0 i ≤ T 2 • kl 0 k ⋆,κ • � K 2 -kχk • K 4 K 3 � ,
and this gives

kχk ≤ K 2 • K 3 K 4 .
This concludes the proof of the theorem. � Remark 3.4. i) The above arguments provide an alternative proof for Theorem 4.2 in [START_REF] Schmitt | Global boundedness for semistable decorated principal bundles with special regard to quiver sheaves[END_REF] for the structure group GL n (C).

ii) Let us use the conventions of the proof of Theorem 3.3. Suppose ((U, P), ϕ) is an O X (m)-twisted affine σ-bump. The set S associated with it in ( 10) is called the generic set of sets of states of the affine bump. Since there are only finitely many possibilities for S, we may fix the generic set of sets of states for our discussion. The notion of χ-(semi)stability of a point in R or R K depends only on its set of sets of states. So, it makes sense to define

K ss (S) := � ψ ∈ N | S is ψ-semistable � .
In [START_REF] Schmitt | Global boundedness for semistable decorated principal bundles with special regard to quiver sheaves[END_REF], p. 481ff, we explained how to deal with the case that K ss (S) 6 = {0}. The main point is that there is a decomposition of K ss (S) into finitely many polyhedral subcones, such that the conditions of ψ-stability and -semistability for the set of sets of states S are constant within the relative interior of each of these cones. In each of these relative interiors, we pick an integral element. Let χ 1 , ..., χ t be the resulting characters of G. In addition, we need to fix for χ j a positive integer N j , satisfying certain requirements, j ∈

[t]. Now, set � ρ j := S Nj (ρ) ⊗ χ j , j ∈ [t].
It is possible to reduce the problem to Theorem 3.3 for bumps attached to the representations � ρ 1 , ..., � ρ t . This will enable us to go from the case of generically totally unstable quiver sheaves to arbitrary quiver sheaves.

Conclusion of the proof of the main theorem

We need to introduce some more notation. Let s ≥ 1 be a natural number and

= � (n j i , i ∈ [w]), j ∈ [s]
� a collection of tuples of integers, such that

• 0 ≤ n j i ≤ n j+1 i ≤ n i , j ∈ [s -1], i ∈ [w], • 0 < w � i=1 n j i < w � i=1 n j+1 i < w � i=1 n i , j ∈ [s -1]. Pick a tuple κ = (κ i , i ∈ [w]) ∈ (R ≥0
) ×w of non-negative real numbers and define further

• n 0 i := 0, n s+1 i := n i , i ∈ [w],
•

n j κ := w � i=1 κ i • (n j i -n j-1 i ), j ∈ [s + 1],
• n 0 κ := 0. Next, set

R s+1 sasc := � δ = (δ 1 , ..., δ s+1 ) ∈ R s+1 � � δ 1 < • • • < δ s+1 � , R s+1 asc := � δ = (δ 1 , ..., δ s+1 ) ∈ R s+1 � � δ 1 ≤ • • • ≤ δ s+1 � , and
OP ,κ := � (δ 1 , ..., δ s+1 ) ∈ R s+1 sasc � � s+1 � j=1 (n j κ -n j-1 κ ) • δ j = 0 � , OP ,κ := � (δ 1 , ..., δ s+1 ) ∈ R s+1 asc � � s+1 � j=1 (n j κ -n j-1 κ ) • δ j = 0 � .
Finally, we introduce

ΩΠ := � (ν, δ) ∈ R w + × R s+1 sasc � � δ ∈ OP ,ν � , ΩΠ := � (ν, δ) ∈ (R ≥0 ) ×w × R s+1 asc � � δ ∈ OP ,ν � .
In our arguments, the map 9,10 ev :

ΩΠ × N -→ R (ν, δ, ψ) 7 -→ w � i=1 s � j=1 � ψ i • (δ j+1 -δ j ) • n j i �
will play an important rôle. Suppose B ⊂ N is a compact subset. For (ν , δ) ∈ ΩΠ , set

�(ν, δ) := max � ev(ν, δ, ψ) � � ψ ∈ B � .
Then,

� : ΩΠ -→ R (ν, δ) 7 -→ �(ν, δ)
is a continuous function ( [START_REF] Berge | Topological spaces, including a treatment of multi-valued functions, vector spaces and convexity[END_REF], p. 116). Denoting the maximum norm on R s+1 and R w by k • k, we set

ΩΠΞ := � (ν, δ) ∈ ΩΠ � � kνk = 1, kδk = 1 � , ΩΠΞ := � (ν, δ) ∈ ΩΠ � � kνk = 1, kδk = 1 � .
The function � admits a maximum on ΩΠΞ , call it C 2 .

Recall that we are assuming that ρ is a homogeneous representation of degree zero. Set W := C |n| . According to [START_REF] Schmitt | Moduli for decorated tuples of sheaves and representation spaces for quivers[END_REF] For this reason, we will assume ρ = ρ a,b,c for the rest of this section.

Next, let us look at a [w]-split vector bundle (E i , i ∈ [w]) on U and a filtration 11 This defines the tuple

E • : 0 ( (E 1 i , i ∈ [w]) ( • • • ( (E s i , i ∈ [w]) ( (E i , i ∈ [w]) of (E i , i ∈ [w]).
(E • ) = � (rk(E j i ), i ∈ [w]), j ∈ [s] � ,
satisfying the conditions stated above. Suppose that

E := w � i=1 E i
is endowed with a non-zero tensor field

ψ : E a,b = (E ⊗a ) ⊕b -→ det(E) ⊗c ⊗ O X (m) |U .
We write

ψ = ψ 1 + • • • + ψ b with ψ β : E ⊗a -→ det(E) ⊗c ⊗ � O X (m) � |U , β ∈ [b]. For i 0 ∈ [w],
we let ι i0 : E i0 -→ E be the obvious inclusion map, and, for β ∈ [b] and a tuple i = (i 1 , ..., i a ) with i α ∈ [w], α ∈ [a], we define

ψ β i := ψ β • (ι i1 ⊗ • • • ⊗ ι ia ). Next, suppose γ = (γ 1 , ..., γ s+1 ) ∈ OP (E • ) . Then, setting E j := � i∈[w] E j i , j ∈ [s + 1],
µ(E • , γ, ψ)

:= -min � γ j1 + • • • + γ ja � � j = (j 1 , ..., j a ) ∈ [s + 1] ×a : ψ |(E j 1 ⊗•••⊗E ja ) ⊕b 6 ≡ 0 � = -min � γ j1 + • • • + γ ja � � j = (j 1 , ..., j a ) ∈ [s + 1] ×a : ∃β ∈ [b] : ψ β |E j 1 ⊗•••⊗E ja 6 ≡ 0 � (12) = -min � γ j1 + • • • + γ ja � � j = (j 1 , ..., j a ) ∈ [s + 1] ×a : ∃β ∈ [b] : ∃i = (i 1 , ..., i a ) ∈ [w] ×a : ψ β i|E j 1 i 1 ⊗•••⊗E ja ia 6 ≡ 0 � .
11 This means that, on a big open subset U ′ ⊂ U , E j i|U ′ will be a subbundle of

E j+1 i|U ′ , E s+1 i := E i , i ∈ [w], j ∈ [s].
Remark 4.1. The function ΩΠΞ (E • ) -→ R, (ν, δ) 7 -→ µ(E • , δ, ψ), is piecewise linear and, therefore, continuous. It depends only on the set

� (β, i, j) ∈ [b] × [w] ×a × [s + 1] ×a � � ψ β i|E j 1 i 1 ⊗•••⊗E ja ia 6 ≡ 0 � .
Having fixed, w, a, b, n, and , there are only finitely many possibilities for this set. Also, given n, the discrete invariant can admit only finitely many values. So, the function δ 7 -→ µ(E • , δ, ψ) belongs to a finite set of continuous functions. Let { Φ 1 , ..., Φ t } be this set of functions.

To conclude the preparations, we need to discuss the Harder-Narasimhan filtration. Let κ ∈ R w + be a tuple of positive real numbers. In the all the notions related to slope semistability, κ will stand for (κ, 0). As explained in [START_REF] Schmitt | Moduli for decorated tuples of sheaves and representation spaces for quivers[END_REF]

µ(E j ) = µ κ � E i , i ∈ [w] � , j ∈ [w]. Now, any [w]-split sheaf (E i , i ∈ [w]
) possesses a Harder-Narasimhan filtration

E • : 0 ( (E 1 i , i ∈ [w]) ( • • • ( (E s i , i ∈ [w]) ( (E i , i ∈ [w]
) with respect to the stability parameter κ. For i ∈ [w], it induces the filtration

E • i : 0 ⊆ E 1 i ⊆ • • • ⊆ E s i ⊆ E i . After removing improper inclusions, we get � E • i : 0 ( � E i 1 ( • • • ( � E i si ( E i .
By our previous remark, this is the Harder-Narasimhan

filtration of E i , i ∈ [w].
Conversely, it is now clear how the Harder-Narasimhan filtration of (E i , i ∈ [w]) is built from the Harder-Narasimhan filtrations of E 1 , ..., E w . In particular,

� µ κ � E 1 i , i ∈ [w] � , ..., µ κ � E i /E s i , i ∈ [w] � � = � µ( � E 1 1 ), ..., µ(E 1 / � E s1 1 ), ..., µ( � E 1 w ), ..., µ(E w / � E sw w )
� .

We see that the Harder-Narasimhan filtration of (E i , i ∈ [w]) does not depend on κ.

4.1. The case of totally unstable quiver sheaves. Note that, by construction, ψ is generically semistable. This is equivalent to the fact that, for every filtration

E • of (E i , i ∈ [w]) and every tuple γ ∈ OP (E • ),κ , 13 µ(E • , γ, ψ) ≥ 0. For χ ∈ N , (E i , i ∈ [w], ψ) will be (κ, χ)-slope semistable, if, for every filtration E • of (E i , i ∈ [w]) and every tuple γ ∈ OP (E • ),κ , satisfying µ(E • , γ, ψ) = 0, the inequality L(E • , γ) + ev (κ, γ, χ) ≥ 0, with L(E • , γ) := s � j=1 � γ j+1 -γ j rk κ (E i , i ∈ [w]) � • � deg κ (E i , i ∈ [w]) • rk κ (E j i , i ∈ [w]) - -deg κ (E j i , i ∈ [w]) • rk κ (E i , i ∈ [w])
� , holds true.

Theorem 4.2. Let the situation be as in Theorem 3.3. Then, there is a constant C 3 , such that, for every κ ∈ Ξ, every χ ∈ N with χ = 0 or χ/kχk ∈ Ω ′ , and every (κ, χ)-slope semistable totally Ω ′ -unstable O X (m)-twisted affine σ-bump ((U, P), ψ) in which (U, P) has type (n, d), the associated [w]-split vector bundle

(E i , i ∈ [w]) satisfies µ max (E i ) ≤ C 3 , i ∈ [w].
Proof. We introduce one more piece of notation which we will use in the proof. For κ = (κ 1 , ..., κ w ) ∈ (R ≥0 ) ×w \ {0}, we set

n(κ) := κ 1 • n 1 + • • • + κ w • n w , d(κ) := κ 1 • d 1 + • • • + κ w • d w , µ(κ) := d(κ) n(κ) .
Let κ 0 ∈ R w + , χ 0 ∈ N with χ 0 = 0 or χ 0 /kχ 0 k ∈ Ω ′ be stability parameters, ((U, P), ψ) a (κ 0 , χ 0 )-slope semistable totally Ω ′ -unstable O X (m)-twisted affine σbump in which (U, P) has type (n, d), and (E i , i ∈ [w]) the associated [w]-split vector bundle on U . Using the notation from above, we assume that E • is the Harder-Narasimhan filtration of (E i , i ∈ [w]) with respect to the stability parameter κ 0 . Then, by [START_REF] Andreas | Vector bundles in algebraic geometry and mathematical physics[END_REF], claim in the proof of Proposition 2.5.2,

∀γ ∈ OP (E • ),κ 0 \ {0} : L(E • , γ) < 0.
It follows that, for every γ ∈ OP (E • ),κ 0 with µ(E • , γ, ψ) = 0, ev(κ 0 , γ, χ 0 ) > 0.

Using the constant C 1 from Theorem 3.3, we introduce the compact set

B := � ψ ∈ N � � � ψ = 0 ∨ ψ kψk ∈ Ω ′ , kψk ≤ C 1 � .
Then, it follows that �(κ 0 , γ) > 0.

Let τ 0 ∈ [t] be such that (ν, δ) 7 -→ µ(E • , δ, ψ) agrees with the function Φ τ0 . We form the continuous function ( [START_REF] Berge | Topological spaces, including a treatment of multi-valued functions, vector spaces and convexity[END_REF], p. 116)

f τ0 : ΩΠ (E • ) -→ R (ν, δ) 7 -→ �(ν, δ) + Φ τ0 (ν, δ).
Note that f τ0 has the property

∀κ ∈ (R ≥0 ) ×s ∀γ ∈ R asc \ {0} : f τ0 (κ, γ) = kγk • f τ0 � κ, γ kγk � . By definition of E • , min � f τ0 (κ 0 , δ) � � δ ∈ OP (E • ),κ 0 : kδk = 1 � > 0.
We introduce the continuous function ( [START_REF] Berge | Topological spaces, including a treatment of multi-valued functions, vector spaces and convexity[END_REF], p. 115)

F τ0 : (R ≥0 ) ×w -→ R ν 7 -→ min � f τ0 (ν, δ) � � δ ∈ OP (E • ),ν : kδk = 1 � . The image of Ξ := { ν ∈ (R ≥0 ) ×w | kνk = 1 } under F is a compact set in
∈ R w + . For κ ∈ (R ≥0 ) ×w , set µ j κ := µ κ � E j i , i ∈ [w] � , j ∈ [s -1]. Note that γ 0 := � µ(κ ′ 0 ) -µ 1 κ ′ 0 , ..., µ(κ ′ 0 ) -µ s+1 κ ′ 0 � ∈ OP (E • ),κ ′ 0 .
We need an upper bound for f τ0 (κ ′ 0 , γ 0 ). We use Formula [START_REF] Hille | Homogeneous vector bundles and Koszul algebras[END_REF]. Let β ∈ [b], i = (i 1 , ..., i a ) ∈ [w] ×a , j = (j 1 , ..., j a ) ∈ [s + 1] ×a be data which compute µ(E • , γ 0 , ψ). Then, we have a non-zero map

E j1 i1 ⊗ • • • ⊗ E ja ia -→ det(E) ⊗c ⊗ O X (m) |U . It follows that µ min (E j1 i1 ) + • • • + µ min (E ja ia ) = µ min (E j1 i1 ⊗ • • • ⊗ E ja ia ) ≤ c • |d| + deg � O X (m) � .
For l ∈ [a], let j ′ l ∈ [s + 1] be the first index with E j l i l = E j ′ l i l . The comments on the Harder-Narasimhan filtration made just before Section 4.1 show [START_REF] Hitchin | The self-duality equations on a Riemann surface[END_REF] µ min (E j l i l ) = µ

j ′ l κ ′ 0 ≥ µ j l κ ′ 0 , l ∈ [a].
We find

f τ0 (κ ′ 0 , γ 0 ) ≤ max � C 2 , c • |d| + deg � O X (m) � -a • µ(κ ′ 0 )
� .

Note that ν 7 -→ µ(ν) is a continuous function on (R ≥0 ) ×w \ {0}. It admits a minimum C 4 on the set Ξ. So,

f τ0 (κ ′ 0 , γ 0 ) ≤ C 5 := max � C 2 , c • |d| + deg � O X (m) � -a • C 4 � .
The constant C 5 depends only on the input data. Altogether we find

C 5 ≥ f τ0 (κ ′ 0 , γ 0 ) = kγ 0 k • f τ0 � κ ′ 0 , γ 0 kγ 0 k � ≥ kγ 0 k • M τ0 2 ,
i.e.,

kγ 0 k ≤ 2 • C 5 M τ0 .
This means

µ 1 κ ′ 0 -µ(κ ′ 0 ) ≤ 2 • C 5 M τ0 or µ(κ ′ 0 ) -µ s+1 ν ′ 0 ≤ 2 • C 5 M τ0 .
Recall that ν 7 -→ µ(ν) admits the minimal value C 4 on the set Ξ. It also attains a maximal value C 6 on this set. The first inequality and the remarks on the Harder-Narasimhan filtration before Section 4.1 give

µ max (E) = µ 1 κ ′ 0 ≤ 2 • C 5 M τ0 + C 6
and the second one

µ min (E) = µ s+1 κ ′ 0 ≥ - 2 • C 5 M τ0 + C 4 .
Both inequalities show that the maximal slope of E is bounded from above. � 4.2. The remaining case. We return to the setting of Remark 3.4, ii). The notation and basic constructions are explained in [START_REF] Schmitt | Global boundedness for semistable decorated principal bundles with special regard to quiver sheaves[END_REF], p. 481ff. Pick a potential generic set of sets of states S. There are three situations to consider: 1) The stability parameter χ satisfies χ/kχk ∈ Ω ′ ; 14 2) χ ∈ K ss (S); 3) χ 6 ∈ K ss (S) and ϑ(χ) > 0.

In the first case, we look at totally Ω ′ -unstable O X (m)-twisted affine σ-bumps which are (κ, χ)-semistable with respect to some parameters κ ∈ Ξ, χ ∈ R >0 • Ω ′ . Here, we may directly apply Theorem 3.3 and 4.2.

Let us speak about the second case. Fix κ ∈ Ξ, χ ∈ K ss (S), and let (U, P, ψ) be a (κ, χ)-semistable O X (m)-twisted affine ρ-bump. It satisfies the condition L κ (E • , α)(≥)0, for every λ ∈ X ⋆,R (D) κ-SL and every reduction β : U ′ -→ P |U ′ /Q G (λ) with µ ρ (β, ψ) = 0 and hχ, λi = 0. This condition does not change, if we replace χ by ℓ • χ, for some positive real number ℓ. So, we may fix any constant C 7 and assume kχk < C 7 . We see that the arguments in the proof of Theorem 4.2 may be adapted to cover this case.

For the third case, pick a face F of the cone K ss (S) and an integral character χ F in the relative interior of that face. Then, there are a new twisting line bundle L F := O X (N F • m) ⊗ L ∨ χF , a closed subset Ω F ⊂ Ω, 15 and a certain representation � ρ F . The latter is homogeneous of degree zero. We need to study totally Ω F -unstable L F -twisted affine � ρ F -bumps which are (κ, χ)-semistable with respect to parameters κ ∈ Ξ and χ ∈ R >0 • Ω F . Theorem 3.3 and 4.2 apply to that situation. � Example 4.3. Let w ≥ 2 be a natural number and � A w = ([w], A, t, h) the quiver whose arrow set is A = { a 1 , ..., a w }, t(a i ) = i, and h(a i ) = ı + 1, i ∈ [w] (Figure 1). Here ı = i, i ∈ [w -1], w + 1 = 1.

14 defined in loc. cit. 15 Let us take the opportunity to correct some notation in that article. On Page 481, second line from below, we should set γ := αα 0 , and, on Page 483, line 11,

Ω F := � β ∈ N F � � kβk = 1 � . � 4 a 3 � 3 a 2 � 2 a 1 � 1 a w � w a w-1 � w -1 a w-2 � w -2
Figure 1. A circular quiver Koike computed the semi-invariants for this quiver [START_REF] Koike | Relative invariants of the polynomial rings over type Ar, � Ar-quivers[END_REF]. His result implies that, for a given dimension vector n = (n i , i ∈ [w]), a representation (f ai , i ∈ [w]) ∈ Rep( � A w , n) is totally unstable if and only if, for all i, j ∈ [w], the homomorphism f : C ni -→ C nj associated with the shortest non-constant path from i to j is not an isomorphism. Note that this example includes the quiver A w depicted in Figure 2.

  are the scalar extensions of the pairings discussed before. The norm on X ⋆,R (D) and X R (D) induced by (•, •) ⋆,R and (•, •) R will be denoted by k • k ⋆ and k • k, respectively. For a real cocharacter l = n � i=1 χ i • e i ∈ X ⋆,R (D), we define the dual real character l

6

 6 because σ is homogeneous of degree zero. Furthermore, χ ∈ N gives hχ, li R = 0. If l is the standard generator of l ∈ X ⋆,R (Λ), we get the condition w � i=1 κ i • deg(E i ) ≥ 0. In the same manner, -l and β give w � i=1 κ i • deg(E i ) ≤ 0. So, we obtain the topological restriction w � i=1 κ i • deg(E i ) = 0. Omitting X ⋆,R (Λ) just means omitting this topological restriction.

  , Proposition 3.1.2, [25], Proposition 2.5.1.2, we may find positive integers a, b, c, subject to the condition a = |n| • c, such that ρ is a direct summand of the natural representation ρ a,b,c of GL n (C) on W a,b,c := (W ⊗a ) ⊕b ⊗

A w : 1 a2-

 1

Figure 2 .

 2 Figure 2. A linear quiver

  , Remark 3.3.2, [5], Proposition 2.4.2, a [w]-split sheaf (E i , i ∈ [w]) is κ-slope semistable 12 if and only if E j , j ∈ [w], is a slope semistable sheaf and

  R. By what we have just observed, it admits a positive maximum M τ0 . This constant depends only on the input data. The preimage of the set (M τ0 /2, ∞) under F |Ξ is a non-empty open subset of Ξ. So, it contains an element κ ′ 0

This means that the closed subset Z := X \ U has codimension at least two in X.

It could, however, be applied when κ is fixed in order to give a simpler proof of the results[START_REF] Schmitt | Global boundedness for semistable decorated principal bundles with special regard to quiver sheaves[END_REF], at least for G = GLn[C).

see Equation[START_REF] Álvarez-Cónsul | Some results on the moduli spaces of quiver bundles[END_REF]

The supremum exists as a real number, because the function ν 7 -→ kλ i k⋆,ν can be extended to the closure Ξ (see Section 1).

The exercise does not depend on the fact that G is semisimple, but rather on the fact that the image of κ, corresponding to ι, here, is contained in SL(W ).

For the definition of µσ(β, ϕ), we refer to[START_REF] Schmitt | Geometric invariant theory and decorated principal bundles[END_REF], Page 347, or[START_REF] Schmitt | Global boundedness for semistable decorated principal bundles with special regard to quiver sheaves[END_REF], Page 449.

This point is not essential for our argument. In general, we may pass to a finite extension of the function field of X.

In Equation[START_REF] Greb | Variation of Gieseker-Maruyama moduli spaces via quiver GIT[END_REF], we will see that there is always a lower bound.

The vector space N has been defined in[START_REF] Bradlow | Special metrics and stability for holomorphic bundles with global sections[END_REF].

See[START_REF] Schmitt | Moduli for decorated tuples of sheaves and representation spaces for quivers[END_REF], Remark 3.4.3, [26], Remark 2.1, ii).

Observe that slope semistability is defined for vector bundles on big open subsets of X.

Acknowledgments. This research was supported by SFB 647 "Space-Time-Matter", project C3 "Algebraic Geometry: Deformations, Moduli and Vector Bundles". My special thanks go to Nikolai Beck for a careful reading of the manuscript and the correction of a mistake.


Modesar Shakoor
email: modesar.shakoor@northwestern.edu

Marc Bernacki
email: marc.bernacki@mines-paristech.fr

Pierre-Olivier Bouchard
email: pierre-olivier.bouchard@mines-paristech.fr

Three-dimensional numerical modeling of ductile fracture mechanisms at the microscale

Keywords: Ductile fracture, damage, micromechanics, remeshing, interfaces, levelset

ABSTRACT

The present work aims at a better understanding and modeling of ductile fracture during the forming of metallic materials. These materials are formed using series of thermomechanical loads where many parameters such as loading type and direction vary, and predictive numerical tools are necessary to understand the occurrence of failure and optimize production costs.

Ductile fracture in metallic materials is the result of voids nucleation, growth, and coalescence mechanisms, leading to a progressive loss of load carrying capacity, until failure [START_REF] Pineau | Failure of metals I: Brittle and ductile fracture[END_REF]. In this work, a micromechanical approach is developed in order to conduct realistic Finite Element (FE) simulations of ductile fracture at the microscale, accounting explicitly for the microstructure of the studied materials. Such simulations require a robust methodology to discretize the microstructure, and then update this discretization during its deformation and progressive failure. Level-Set (LS) functions are used to represent all interfaces [START_REF] Osher | Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations[END_REF], and ease the modeling of void linkage events, that are difficult to handle with standard FE methods. This implicit representation of interfaces raises multiple issues. First, discontinuous material behavior at interfaces remains to be modeled. Second, large plastic strains can only be reached using several remeshing operations, which induce an important diffusion of the volume and morphology of the microstructure when the latter is carried only by LS functions. A mesh generation method is developed to mesh interfaces with a conforming FE discretization, hence explicitly representing discontinuities in material behavior. The quality and accuracy of this discretization is maintained during deformation thanks to a mesh adaption algorithm enhanced with a volume conservation constraint [START_REF] Shakoor | An efficient and parallel level set reinitialization method Application to micromechanics and microstructural evolutions[END_REF]. Additionally, real microstructures typically feature a complex morphology that requires a very fine discretization. Using a local error estimator based on LS functions, the discretization is automatically refined where necessary [START_REF] Quan | Anisotropic mesh adaptation with optimal convergence for finite elements using embedded geometries[END_REF]. This error estimator relies on the geometric properties of LS functions, which are maintained during deformation thanks to a new direct LS reinitialization technique [START_REF] Shakoor | An adaptive level-set method with enhanced volume conservation for simulations in multiphase domains[END_REF]. Finally, a computational fracture mechanics approach and stress-based fracture criteria are proposed to model the nucleation of microscopic voids. The interest of these numerical developments and micromechanical models is first demonstrated at the scale of representative volume elements with statistically generated microstructures. Then, simulations of real microstructures with measured boundary conditions are addressed. Three-dimensional X-ray images are meshed and displacement fields measured by image correlation are applied at the boundaries of the FE domain in order to conduct simulations in conditions that are as close as possible to what is observed during in-situ experiments [START_REF] Buljac | Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations[END_REF]. Such massive 3D simulations can be achieved thanks to the efficiency of all numerical developments performed in this work, and their implementation in a distributed computing framework. Error measurements accounting for the volume and shape of each void prove the validity of these results, and the interest of this numerical validation framework with respect to standard micromechanical approaches [START_REF] Shakoor | On the choice of boundary conditions for micromechanical simulations based on 3D imaging[END_REF].