The Harder–Narasimhan stratification of the moduli stack of $$G$$G-bundles via Drinfeld’s compactifications
By Simon Schieder. 2012
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We use Drinfeld’s relative compactifications $${\overline{\mathop {{\mathrm{Bun}}}\nolimits }}_P$$Bun¯P and the Tannakian viewpoint on principal bundles to construct the Harder–Narasimhan stratification of the moduli stack $$\mathop {{\mathrm{Bun}}}\nolimits _G$$BunG of $$G$$G-bundles on an algebraic curve in arbitrary characteristic, generalizing the stratification for $$G=\mathop {\mathrm{GL}}\nolimits _n$$G=GLn due to Harder and Narasimhan to the case of an arbitrary reductive group $$G$$G. To establish the stratification on the set-theoretic level, we exploit a Tannakian interpretation of the Bruhat decomposition and give a new and purely geometric proof of the existence and uniqueness of the canonical reduction in arbitrary characteristic. We furthermore provide a Tannakian interpretation of the canonical reduction in characteristic $$0$$0 which allows to study its behavior in families. The substack structures on the strata are defined directly in terms of Drinfeld’s compactifications $${\overline{\mathop {{\mathrm{Bun}}}\nolimits }}_P$$Bun¯P, which we generalize to the case where the derived group of $$G$$G is not necessarily simply connected. Using $${\overline{\mathop {{\mathrm{Bun}}}\nolimits }}_P$$Bun¯P, we establish various properties of the stratification, including finer information about the structure of the individual strata and a simple description of the strata closures.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/035a65cfeb84137cc6cb7ec012b557bab5319b7b
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Weil's Conjecture for Function Fields (2019, methodology): “For a proof, we refer the reader to [5] or [47].”