[Paper Review] Associated graded of Hodge modules and categorical sl_2 actions

Sabin Cautis, C. A. F. Dodd+1 more|arXiv (Cornell University)|Mar 24, 2016

Algebraic Geometry and Number Theory28 references3 citations

TL;DR

This paper computes the associated graded of the Hodge module pushforward under an open embedding into a product of complementary Grassmannians, using a categorical sl₂ action on D_h-modules. The key result identifies the associated graded as a pushforward of a line bundle from a locally closed subset of the cotangent bundle, with a natural weight filtration tied to orbit components.

ABSTRACT

One of the most mysterious aspects of Saito's theory of Hodge modules are the Hodge and weight filtrations that accompany the pushforward of a Hodge module under an open embedding. In this paper we consider the open embedding in a product of complementary Grassmannians given by pairs of transverse subspaces. The push-forward of the structure sheaf under this open embedding is an important Hodge module from the viewpoint of geometric representation theory and homological knot invariants. We compute the associated graded of this push-forward with respect to the induced Hodge filtration as well as the resulting weight filtration. The main tool is a categorical $\sl_2$ action on the category of $\D_h$-modules on Grassmannians. Along the way we also clarify the interaction of kernels for $\D_h$-modules with the associated graded functor. Both of these results may be of independent interest.

Motivation & Objective

• To compute the associated graded of the Hodge module pushforward j_*O_U for an open embedding j: U → G(k,N) × G(N−k,N), where U parametrizes transverse pairs of subspaces.
• To understand the structure of the Hodge and weight filtrations induced by this pushforward, especially in the context of geometric representation theory and homological knot invariants.
• To establish a link between categorical sl₂ actions on D_h-modules and Saito’s theory of mixed Hodge modules via the associated graded functor.
• To generalize the computation to cominuscule flag varieties, including Lagrangian Grassmannians and quadrics, and to conjecture analogous results for coherent sheaves on cotangent bundles.

Proposed method

• Utilizes a categorical sl₂ action on the category of D_h-modules on Grassmannians, constructed via kernels supported on conormal varieties of incidence correspondences.
• Applies the associated graded functor gr: D_h-mod → O_{T*X}-mod to relate filtered D-modules to coherent sheaves on cotangent bundles.
• Analyzes the interaction between kernels of D_h-modules and the associated graded functor, clarifying how filtrations behave under pushforward.
• Employs the Riemann-Hilbert correspondence and the faisceaux-fonctions correspondence to relate constructible sheaves to D-modules and Hodge modules.
• Studies the structure of G-orbits on G/P × G/Q for cominuscule parabolic subgroups, identifying a linearly ordered set of orbits Z₀, ..., Z_k with decreasing dimension.
• Conjectures that the associated graded of the Hodge module pushforward is a direct sum of shifted IC-sheaves supported on closures of these orbits, with a filtration indexed by codimension.

Experimental results

Research questions

• RQ1 What is the structure of the associated graded of the Hodge module j_*O_U with respect to the Hodge filtration, where U is the open locus of transverse pairs in a product of complementary Grassmannians?
• RQ2 How does the weight filtration on this associated graded relate to the stratification of the cotangent bundle by G-orbits?
• RQ3 Can the categorical sl₂ action on D_h-modules on Grassmannians be used to compute the associated graded of Hodge module pushforwards?
• RQ4 What is the relationship between the kernels of D_h-modules and the associated graded functor in this context?
• RQ5 Can the results be generalized to other cominuscule flag varieties, such as Lagrangian Grassmannians or even-dimensional quadrics?

Key findings

• The associated graded of the Hodge module j_*O_U is isomorphic to the pushforward of a line bundle from a locally closed subset of T^*X, where X = G(k,N) × G(N−k,N).
• The weight filtration on gr(G(j_*O_U)) is naturally realized as a filtration indexed by the codimension of G-orbits Z_s in X, with gr^W_s being supported on the closure of Z_s.
• The associated graded of the D_h-module underlying the Hodge module is identified as a direct sum of shifted IC-sheaves: gr^W_s(j_*δ_{Z₀,m}) ≅ IC_{Z̄_s,m}{s} for s = 0,…,k.
• The kernel inducing the equivalence between D_h-modules on G/P and G/Q is shown to induce an equivalence on the associated graded side, with the associated graded kernel being a pushforward of a line bundle from a normalization of the conormal variety.
• For cominuscule flag varieties, the irreducible components of T^*G/P ×_g T^*G/Q are the closures of conormal bundles to G-orbits Z_s, and the associated graded kernel is expected to be a direct image of a line bundle from a dense open subset of this variety.
• Conjectures are formulated that the associated graded of the Hodge module pushforward is a direct sum of shifted IC-sheaves, and that the kernel on the O_{T^*X}-side is a pushforward of a line bundle from a normalization of the conormal variety.