Formal orbifolds and orbifold bundles in positive characteristic

Abstract

Algebraic parabolic bundles on smooth projective curves over algebraically closed field of positive characteristic is defined. It is shown that the category of algebraic parabolic bundles is equivalent to the category of orbifold bundles defined in [KP]. Tensor, dual, pullback and pushforward operations are also defined for parabolic bundles.

Ramification theory and formal orbifolds in arbitrary dimension

“…The following analogue of [KP,Lemma 2.12] holds in higher dimension as well with essentially the same proof.…”

Stability of pullback of orbifold bundles

“…So f induces a morphism f : (Y, f * P) −→ (X, P) of formal orbifold curves. By [5,Lemma 2.12], the following hold.…”

“…Let k be an algebraically closed field. The notion of a formal orbifold curve was introduced in [8] when k = C, and was later generalized over fields of arbitrary characteristic in [5]. In this section, we recall the definitions of the formal orbifold curves, the morphisms among them and some properties.…”

“…Roughly, a formal orbifold curve is a pair (X, P) consisting of a smooth k-curve X together with a data given by certain local Galois extensions attached to finitely many points of the curve. Vector bundles on a formal orbifold curve were introduced and studied in [6]. Our first objective is to introduce and study the stability condition for such bundles.…”

“…, m n . It was observed in [6,Proposition 5.15] that the category of vector bundles on (X, P) is equivalent to the category of parabolic vector bundles on X with respect to the divisor 1≤i≤n x i and with rational weights above each point x i that are of the form a/m i for 0 ≤ a < m i . We also have the notion of parabolic slopes and parabolic (semi)stability on the later category.…”