[Paper Review] Koszul duality and mixed Hodge modules

Pramod N. Achar, S. Kitchen|arXiv (Cornell University)|May 11, 2011

Algebraic structures and combinatorial models10 references3 citations

TL;DR

This paper establishes a winnowing-based approach to Koszul duality for mixed Hodge modules on smooth complex varieties with affine even stratifications, proving the winnowed category is Koszul and constructing a canonical degrading functor to perverse sheaves. The key contribution is a formality result generalizing Schnürer’s theorem, resolving the asymmetry in earlier approaches by avoiding arbitrary choices in the subcategory construction used in Beilinson–Ginzburg–Soergel.

ABSTRACT

We prove that on a certain class of smooth complex varieties (those with "affine even stratifications"), the category of mixed Hodge modules is "almost" Koszul: it becomes Koszul after a few unwanted extensions are eliminated. We also give an equivalence between perverse sheaves on such a variety and modules for a certain graded ring, obtaining a formality result as a corollary. For flag varieties, these results were proved earlier by Beilinson-Ginzburg-Soergel using a rather different construction.

Motivation & Objective

• To resolve the asymmetry in Koszul duality constructions for mixed Hodge modules versus $ε$-adic sheaves, where the former required arbitrary choices in subcategory selection.
• To develop a canonical, choice-free alternative to the subcategory approach used in Beilinson–Ginzburg–Soergel for mixed Hodge modules by introducing a 'winnowing' construction that splits extensions.
• To construct a well-behaved degrading functor from the winnowed category of mixed Hodge modules to perverse sheaves, enabling a formality result.
• To show that on full flag varieties, the winnowed category is canonically equivalent to the subcategories constructed in [BGS], thus unifying the approaches.
• To generalize Schnürer’s formality theorem by establishing a canonical autoequivalence relating two Koszul gradings on the category of perverse sheaves.

Proposed method

• Introduce a 'winnowing' construction that adds new morphisms to split previously nonsplit extensions, replacing the subcategory approach used in [BGS] for mixed Hodge modules.
• Define the winnowed category of mixed Hodge modules as the full subcategory of objects whose cohomology vanishes in certain degrees, ensuring Koszulity.
• Construct a degrading functor from the winnowed category of mixed Hodge modules to perverse sheaves using special rat-projective resolutions.
• Use the graded ring $\mathbb{E}$ associated to the endomorphism algebra of the direct sum of simple objects to establish a Koszul duality equivalence.
• Prove that the degrading functor is compatible with the Koszul grading and induces an equivalence between the winnowed category and the category of graded modules over $\mathbb{E}$.
• Establish a canonical autoequivalence $\tilde{\mu}$ on the category of perverse sheaves that relates the two Koszul gradings induced by the rat and $\zeta$ functors.

Experimental results

Research questions

• RQ1 Can Koszul duality for mixed Hodge modules be established without arbitrary choices, unlike the subcategory approach in [BGS]?
• RQ2 Is there a canonical way to 'winnow' the category of mixed Hodge modules to make it Koszul, independent of choices in the construction?
• RQ3 Does the degrading functor from the winnowed category of mixed Hodge modules to perverse sheaves exist and behave well categorically?
• RQ4 How does the winnowed category of mixed Hodge modules relate to the subcategories constructed in [BGS] on full flag varieties?
• RQ5 Can the two Koszul gradings on the category of perverse sheaves—induced by the rat and $\zeta$ functors—be related by a canonical autoequivalence?

Key findings

• The winnowed category of mixed Hodge modules on a variety with an affine even stratification is Koszul, as proven in Theorem 5.7.
• A canonical degrading functor from the winnowed category of mixed Hodge modules to perverse sheaves exists and is constructed in Theorem 6.6.
• On full flag varieties, the winnowed category is canonically equivalent to the subcategories constructed in [BGS], resolving the choice-dependence of the original approach.
• The degrading functor induces a formality result, generalizing a theorem of Schnürer, by relating the two Koszul gradings on perverse sheaves.
• A canonical autoequivalence $\tilde{\mu}$ of the category of perverse sheaves satisfies $\tilde{\mu} \circ \mathrm{rat} \cong \zeta \circ \beta$, relating the two gradings.
• The graded ring $\mathbb{E}$ associated to the endomorphism algebra of the direct sum of simple objects is isomorphic to $E$, and the category of graded $\mathbb{E}$-modules is equivalent to the category of semisimple $\underline{E}$-modules.