[Paper Review] Schematic HN stratification for families of principal bundles and lambda modules

Sudarshan Gurjar, Nitin Nitsure|arXiv (Cornell University)|Aug 28, 2012

Algebraic Geometry and Number Theory5 references3 citations

TL;DR

This paper establishes a schematic Harder-Narasimhan (HN) stratification for families of principal G-bundles on curves in characteristic zero, proving each HN stratum is a locally closed subscheme with a universal property. It further extends this to flat families of pure $Λ$-modules in arbitrary dimensions and mixed characteristic, showing that $Λ$-modules of fixed HN type form an Artin stack.

ABSTRACT

For a family of principal bundles with a reductive structure group on a family of curves in characteristic zero, it is known that the Harder Narasimhan type of its restriction to each fiber varies semicontinuously over the parameter scheme of the family. This defines a stratification of the parameter scheme by locally closed subsets, known as the Harder-Narasimhan stratification. In this note, we show how to endow each Harder-Narasimhan stratum with the structure of a locally closed subscheme of the parameter scheme, which enjoys the universal property that under any base change the pullback family admits a relative Harder-Narasimhan reduction with a given Harder-Narasimhan type if and only if the base change factors through the schematic stratum corresponding to that Harder-Narasimhan type. This has the consequence that principal bundles of a given Harder Narasimhan type form an Artin stack. We also prove a similar result showing the existence of a schematic Harder-Narasimhan filtration for flat families of pure sheaves of $Λ$-modules (in the sense of Simpson) in arbitrary dimensions and in mixed characteristic, generalizing the result for sheaves of ${\mathcal O}$-modules proved earlier by Nitsure. This again has the implication that $Λ$-modules of a fixed Harder-Narasimhan type form an Artin stack.

Motivation & Objective

• To construct a schematic structure on the Harder-Narasimhan strata of a family of principal G-bundles over curves in characteristic zero.
• To prove that each HN stratum admits a universal property: a base change factors through it if and only if the pulled-back family has constant HN type.
• To generalize the schematic HN stratification to flat families of pure $Λ$-modules (in the sense of Simpson) over projective schemes in arbitrary dimensions and mixed characteristic.
• To show that the substack of $Λ$-modules with fixed HN type is an algebraic stack, extending Nitsure's result for $Λ = Ó$.
• To establish boundedness and existence of finite-type atlases for such stacks under suitable conditions.

Proposed method

• Use the canonical reduction of principal G-bundles to define the HN type in the closed positive Weyl chamber $τ \subset \mathbb{Q} \otimes X_*(T)$.
• Construct the schematic stratum $S^\tau(E) \subset S$ as a locally closed subscheme via a moduli-theoretic universal property.
• Apply Hilbert scheme techniques and unramifiedness of the Quot scheme parametrizing HN filtrations to prove the schematic structure.
• Use induction on the length of the HN filtration, reducing the problem to lower-length cases via successive quotients.
• Leverage the existence of a split almost polynomial sheaf of differential operators $Λ$ to define $Λ$-module structures and purity conditions.
• Prove that the relative HN filtration is unique and compatible with base change, ensuring the schematic stratification is well-defined.

Experimental results

Research questions

• RQ1 Can the Harder-Narasimhan stratification of a family of principal G-bundles be endowed with a canonical scheme structure?
• RQ2 Does the schematic HN stratum have a universal property that classifies base changes admitting a relative HN filtration of fixed type?
• RQ3 Can the schematic HN stratification be extended to families of $Λ$-modules in mixed characteristic and arbitrary dimension?
• RQ4 Do $Λ$-modules of fixed HN type form an algebraic stack under suitable flatness and purity conditions?
• RQ5 Under what conditions does the stack of $Λ$-modules of fixed HN type admit a finite-type atlas?

Key findings

• Each Harder-Narasimhan stratum $S^\tau(E)$ in the parameter scheme $S$ of a family of principal G-bundles admits a unique structure of a locally closed subscheme with the desired universal property.
• The schematic HN stratification of $S$ for $Λ$-modules is constructed via a descending induction on the length of the HN filtration, using Quot schemes and unramified morphisms.
• The tangent space to the Quot scheme parametrizing the first HN quotient is trivial, implying the Quot morphism is unramified and hence a closed immersion.
• The stack $\Lambda Coh_{X/S}^\tau$ of $Λ$-modules with fixed HN type $\tau$ is a locally closed substack of the Artin stack $\Lambda Coh_{X/S}$.
• When boundedness holds for semistable $Λ$-modules, each stack $\Lambda Coh_{X/S}^\tau$ admits an atlas of finite type over $S$, generalizing Nitsure's result.
• The construction works in arbitrary characteristic, including mixed characteristic, and applies to families of projective schemes over locally noetherian bases.