[Paper Review] Semistability criterion for parabolic vector bundles on curves

Indranil Biswas, Ajneet Dhillon|arXiv (Cornell University)|Mar 7, 2011

Algebraic Geometry and Number Theory10 references3 citations

TL;DR

This paper establishes a cohomological semistability criterion for parabolic vector bundles on curves with rational weights: a parabolic bundle is semistable if and only if there exists another parabolic bundle such that the cohomologies of their underlying tensor product vanish in all degrees. The key advance is replacing a non-canonical auxiliary bundle in prior work with the trivial line bundle, simplifying the criterion and generalizing Faltings' criterion for ordinary vector bundles.

ABSTRACT

We give a cohomological criterion for a parabolic vector bundle on a curve to be semistable. It says that a parabolic vector bundle $E$ with rational parabolic weights is semistable if and only if there is another parabolic vector bundle $F$ with rational parabolic weights such that the cohomologies of the vector bundle underlying the para- bolic tensor product $E \otimes F$ vanish. This criterion generalizes the known semistability criterion of Faltings for vector bundles on curves and significantly improves the result in [Bis07].

Motivation & Objective

• To generalize Faltings' cohomological semistability criterion for vector bundles to the parabolic setting.
• To eliminate the need for a non-canonical auxiliary parabolic bundle in prior criteria, replacing it with the trivial line bundle.
• To provide a cleaner, more canonical criterion for semistability of parabolic vector bundles with rational parabolic weights.
• To establish the equivalence between semistability and the vanishing of cohomology in all degrees for the tensor product with some parabolic bundle.
• To use root stacks and stack-theoretic methods to derive the criterion in a systematic and geometric way.

Proposed method

• The authors use root stacks to realize parabolic vector bundles as equivariant vector bundles on a Galois cover, enabling the application of stacky cohomological tools.
• They define a tensor category of parabolic bundles with weights in (1/r)ℤ and establish an equivalence between parabolic bundles on X and vector bundles on the root stack X_{D,r}.
• The cohomology vanishing criterion is transferred from the root stack to the base curve using the Leray spectral sequence and the fact that pushforward is exact in characteristic zero.
• The proof leverages the fact that semistable G-linearized bundles on a quotient stack [Y/G] descend to semistable parabolic bundles on X.
• For the case of ℙ¹ with one parabolic point, the result is proven directly via decomposition of the underlying bundle into line bundles.
• The general case uses Faltings' criterion on the Galois cover Y to construct a G-equivariant bundle with vanishing cohomology, which descends to the required parabolic bundle.

Experimental results

Research questions

• RQ1 Can a cohomological criterion for semistability of parabolic vector bundles be formulated that generalizes Faltings' criterion for ordinary vector bundles?
• RQ2 Is it possible to replace the non-canonical auxiliary bundle in prior criteria with a canonical one, such as the trivial line bundle?
• RQ3 How does the theory of root stacks facilitate the translation of parabolic bundles into equivariant bundles on covers?
• RQ4 What conditions ensure that cohomology vanishing of the tensor product implies semistability in the parabolic setting?
• RQ5 Can the semistability of a parabolic bundle be characterized purely by the existence of a dual parabolic bundle such that their tensor product has no cohomology?

Key findings

• A parabolic vector bundle E_* on a curve X with rational parabolic weights is semistable if and only if there exists a parabolic vector bundle F_* such that H^i(X, (E_* ⊗ F_*)₀) = 0 for all i.
• The auxiliary bundle F_* in the criterion can be chosen to be the trivial line bundle O_X with trivial parabolic structure, significantly simplifying the criterion.
• The proof relies on the equivalence between parabolic bundles on X and vector bundles on the root stack X_{D,r}, which allows the use of stacky cohomological tools.
• The semistability of a parabolic bundle is equivalent to the semistability of its associated G-linearized bundle on a Galois cover Y → X.
• The cohomology vanishing criterion is preserved under the pushforward along the morphism π: X_{D,r} → X due to the exactness of π_* in characteristic zero.
• The result improves upon earlier criteria by removing dependence on a non-canonical auxiliary bundle, making the criterion more intrinsic and geometrically natural.