We define a parameter dependent notion of stability for principal bundles with a certain local decoration, which generalizes both parabolic and level structures, and construct their coarse moduli space. A necessary technical step is the construction of the moduli space of tuples of vector bundles with a global and a local decoration, which we call decorated tumps. We introduce a notion of asymptotic stability for decorated tumps and show, that stable decorated principal bundles can be described as asymptotically stable decorated tumps. 1 2 NIKOLAI BECK Notation and Conventions. In this work we will identify a geometric vector bundle E with its sheaf of sections. If F is a subsheaf of E, the subbundle generically generated by F iswhere T is the torsion subsheaf of E/F . We denote by P(E) the hyperplane bundle Proj(Sym * E). For x ∈ R we set [x] + := max{x, 0}. If X and Y are schemes, we denote by pr X and pr Y the canonical projection from the product X × Y to X and Y respectively.Acknowledgement. This article presents the main result of the author's PhD thesis [3]. The author would like thank his advisor A. Schmitt for his guidance.
PreliminariesIn order to fix the notation we recall some facts about split vector bundles.1.1. Split Vector Spaces. Let T be a finite index set.The category of T -split vector spaces is an abelian category. The dimension vector of a T -split vectorof T -split subspaces and α is a k-tuple of positive rational numbers.Given a one-parameter subgroup λ of SL κ T (V ), there are a number k, weights γ i < . . . < γ k+1 and a decomposition ofLet m be the least common denominator of γ 1 , . . . , γ k+1 and define a one-parameter subgroup λ of SL κ T (V ) by λ(c) · v := c mγi v for v ∈ V i , 1 ≤ i ≤ k + 1 and c ∈ C * . Then (V • , mα) is the associated weighted flag of λ Definition 1.4. A representation ρ : GL(d, C) → GL(V ) is homogeneous of degree γ, if ρ(c · id) = c γ id V for all c ∈ C * . Proposition 1.5. Let W be a T -split vector space, κ ∈ Z T >0 and ρ : GL T (W ) → GL(V ) a homogeneous representation of degree γ. Then there are natural numbers a, b, c with γ = a − c dim κ (W ), such that V is a direct summand of the representation W ⊕κ a,b,c := W ⊕κ ⊗a ⊕b ⊗ dimκ(W ) W ⊕κ ⊗−c .Proof. This is Proposition 2.5.1.2 in [15].1.2. Split Vector Bundles. Let X be a smooth projective curve over the complex numbers and T a finite index set.Remark 1.7. The category of T -split sheaves is an abelian category.Definition 1.8. Let F be a T -split sheaf, κ ∈ Q T >0 and χ ∈ Q T . The χ-rank and the (κ, χ)-degree of F arerespectively. If rk χ (F ) = 0, then the (κ, χ)-slope of F isRemark 1.9. Suppose κ is T -tuple of positive integers. Then, the rank of the κ-total sheaf F ⊕κ := t∈T F t ⊕κt is given by rk(F ⊕κ ) = rk κ (F ).Definition 1.10. A T -split vector bundle is a T -split sheaf E = (E t , t ∈ T ), such that for every t ∈ T the sheaf E t is a vector bundle. A morphism of T -split vector bundles is a morphism of T -split sheaves.Remark 1.11. The datum of a T -split vector bundle with rank ve...
