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Modularity of special cycles on Shimura varieties: a survey

François Greer, Salim Tayou

Abstract

We survey recent results on a conjecture of Kudla regarding the modularity of generating series of special cycle classes in toroidal compactifications of orthogonal and unitary Shimura varieties. Along the way, we formulate several conjectures on related phenomena for special cycles in other types of Shimura varieties, as well as on more general quotients of period domains.

Modularity of special cycles on Shimura varieties: a survey

Abstract

We survey recent results on a conjecture of Kudla regarding the modularity of generating series of special cycle classes in toroidal compactifications of orthogonal and unitary Shimura varieties. Along the way, we formulate several conjectures on related phenomena for special cycles in other types of Shimura varieties, as well as on more general quotients of period domains.
Paper Structure (21 sections, 8 theorems, 76 equations, 5 tables)

This paper contains 21 sections, 8 theorems, 76 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Compactifications
  3. Organization of the paper
  4. Acknowledgments
  5. Orthogonal and unitary Shimura varieties
  6. The orthogonal case
  7. The unitary case
  8. PEL Shimura varieties
  9. Siegel modular varieties
  10. Quaternionic Shimura varieties
  11. PEL Shimura varieties
  12. Modularity in compactifications
  13. Satake-Baily-Borel and $L^2$-cohomology
  14. The Boundary Stratification
  15. Toroidal compactification
  16. ...and 6 more sections

Key Result

Theorem 2.1

The generating series $\Theta^g_L$ is the $q$-expansion of a holomorphic modular form of genus $g$ and weight $\frac{n+2}{2}$ with respect to the Siegel modular group $\mathrm{Sp}_{2g}(\mathbb Z)$, that is:

Theorems & Definitions (22)

  • Theorem 2.1
  • Conjecture 2.2
  • Theorem 2.3
  • Conjecture 2.4
  • Conjecture 3.1
  • Conjecture 3.2
  • Conjecture 3.3
  • Definition 3.4
  • Remark 3.4
  • Definition 3.5
  • ...and 12 more