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Coherent cohomology of Shimura varieties and automorphic forms

Jun Su

Abstract

We show that the cohomology of canonical extensions of automorphic vector bundles over toroidal compactifications of Shimura varieties can be computed by relative Lie algebra cohomology of automorphic forms. Our result is inspired by and parallel to Borel and Franke's work on the cohomology of automorphic local systems on locally symmetric spaces, and also generalizes a theorem of Mumford.

Coherent cohomology of Shimura varieties and automorphic forms

Abstract

We show that the cohomology of canonical extensions of automorphic vector bundles over toroidal compactifications of Shimura varieties can be computed by relative Lie algebra cohomology of automorphic forms. Our result is inspired by and parallel to Borel and Franke's work on the cohomology of automorphic local systems on locally symmetric spaces, and also generalizes a theorem of Mumford.

Paper Structure

This paper contains 27 sections, 68 theorems, 346 equations.

Table of Contents

  1. Coherent cohomology of Shimura varieties
  2. Automorphic vector bundles
  3. Automorphic vector bundles as sheaves
  4. Cohomology of $\widetilde{V}$
  5. Toroidal compactifications
  6. $\sigma$-neighborhoods and $\sigma$-coordinates
  7. Toroidal compactifications of $\mathrm{Sh}_\mathbb{K}$
  8. Canonical extensions of automorphic vector bundles
  9. The sheaf on SNC toroidal compactifications
  10. The sheaves and
  11. Growth conditions
  12. Definition of the sheaves
  13. $(\phi_\sigma^*\mathcal{C}^\infty_{\mathrm{mg}})|_\Omega$ and $(\phi_\sigma^*\mathcal{A}^{0,i}_{\mathrm{dmg}})|_\Omega$
  14. Proof of Proposition \ref{['dmglocal']}
  15. Identification of $(\phi_\sigma^*\mathcal{C}^\infty_{\mathrm{mg}})|_\Omega$ and $(\underline{z}^*\mathcal{C}^\infty_{\mathrm{lg},r})|_\Omega$
  16. ...and 12 more sections

Key Result

Theorem 1

Let $\widetilde{V}$ be an automorphic vector bundle over $\mathrm{Sh}_\mathbb{K}$ arising from a representation $V$ of $P_h$, $\widetilde{V}^{\mathrm{can}}$ be its canonical extension over an admissible toroidal compactification $\mathrm{Sh}_{\mathbb{K},\Sigma}$ of $\mathrm{Sh}_\mathbb{K}$, then the

Theorems & Definitions (145)

  • Theorem
  • Example 1.1
  • Proposition 1.2
  • Lemma 1.3
  • proof
  • Definition 1.4
  • Definition 1.5
  • Example 1.6
  • Proposition 1.7
  • Definition 2.1
  • ...and 135 more