Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Corank spectral sequence for locally symmetric varieties

Shouhei Ma

TL;DR

This work introduces a corank-based spectral sequence to compute the weight-graded edge pieces Gr^{W}_{2n-p}F^{n}H^{2n-p-*}(X) of the mixed Hodge structure on cohomology of locally symmetric varieties X = {\mathcal D}/{\Gamma}. The E^{1}-page is explicitly given by cusp cohomology ⊕_{F∈C(i)} H^{n(i)-i-j}(Γ_{F,ℓ}, H^{0}(Ω^{p}_{\overline{Y}_{F}})^{*}) and converges to Gr^{W}_{2n-p}F^{n}H^{2n-p-m}(X), with a degeneracy range that yields concrete descriptions of edge pieces. The theory unifies Hodge-theoretic edge data with boundary geometry via toroidal compactifications, cellular cosheaves, and boundary cusp groups, and yields an Euler-number identity connecting open and boundary contributions. Specializations to Hilbert, Siegel, and orthogonal modular varieties demonstrate how cusp forms and cusp-cohomology govern the high-weight Hodge components, providing new, purely Hodge-theoretic proofs and computational tools independent of Eisenstein series techniques.

Abstract

We construct a new type of spectral sequences for the mixed Hodge structures on the cohomology of locally symmetric varieties. These spectral sequences converge to the edge components in the Hodge triangles, and the E1-terms are expressed by group cohomology associated to the cusps. They already degenerate at E1 in a certain range, which gives a simple expression of some Hodge components. An identity of holomorphic Euler numbers is obtained as a consequence.

Corank spectral sequence for locally symmetric varieties

TL;DR

This work introduces a corank-based spectral sequence to compute the weight-graded edge pieces Gr^{W}_{2n-p}F^{n}H^{2n-p-*}(X) of the mixed Hodge structure on cohomology of locally symmetric varieties X = {\mathcal D}/{\Gamma}. The E^{1}-page is explicitly given by cusp cohomology ⊕_{F∈C(i)} H^{n(i)-i-j}(Γ_{F,ℓ}, H^{0}(Ω^{p}_{\overline{Y}_{F}})^{*}) and converges to Gr^{W}_{2n-p}F^{n}H^{2n-p-m}(X), with a degeneracy range that yields concrete descriptions of edge pieces. The theory unifies Hodge-theoretic edge data with boundary geometry via toroidal compactifications, cellular cosheaves, and boundary cusp groups, and yields an Euler-number identity connecting open and boundary contributions. Specializations to Hilbert, Siegel, and orthogonal modular varieties demonstrate how cusp forms and cusp-cohomology govern the high-weight Hodge components, providing new, purely Hodge-theoretic proofs and computational tools independent of Eisenstein series techniques.

Abstract

We construct a new type of spectral sequences for the mixed Hodge structures on the cohomology of locally symmetric varieties. These spectral sequences converge to the edge components in the Hodge triangles, and the E1-terms are expressed by group cohomology associated to the cusps. They already degenerate at E1 in a certain range, which gives a simple expression of some Hodge components. An identity of holomorphic Euler numbers is obtained as a consequence.
Paper Structure (29 sections, 27 theorems, 129 equations, 4 figures)

This paper contains 29 sections, 27 theorems, 129 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Mixed Hodge theory
  3. Weight spectral sequence
  4. Interpretation via cellular cosheaves
  5. Toroidal compactifications
  6. Baily-Borel compactification
  7. Stabilizer of a cusp
  8. Siegel domain realization
  9. Toroidal compactification
  10. Structure of the boundary
  11. The corank spectral sequences
  12. Corank filtration
  13. The $E^1$ page
  14. Acyclicity
  15. Complements
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $0\leq p <n$ be fixed. Then there exists a first-quadrant homological spectral sequence where the last isomorphism is valid for $m \geq 2$. We have $E^{1}_{i,j}\ne 0$ only in the range where $0\leq d(p)\leq r$ is the largest corank with $n(d(p))\leq n-p$.

Figures (4)

  • Figure 1: ${\rm Gr}^{W}_{2n-p}F^{n}H^{2n-p-m}(X)$
  • Figure 2: $E^1$ page
  • Figure 3: Range of $E^1$ degeneration
  • Figure 4: $(\Delta^{+}, \partial\Delta^{+})$

Theorems & Definitions (62)

  • Theorem 1.1: Theorem \ref{['thm: main']}
  • Corollary 1.2: § \ref{['ssec: Euler number']}
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Theorem 4.1
  • Remark 4.2
  • Lemma 4.3
  • ...and 52 more