Correspondence of Hermitian modular forms to cycles associated to \(\mathrm{SU}(p,2)\)
type: Item
instance of: scholarly article
title: Correspondence of Hermitian modular forms to cycles associated to \(\mathrm{SU}(p,2)\)
zbMATH Open document ID: 0559.10027
author: Yue Lin L. Tong
author: S. P. Wang
published in: Journal of Differential Geometry
publication date: 1983-01-01T00:00:00Z
review text: This paper is one of a series of papers by the authors concerning a fascinating relation between certain families of cycles, defined by totally geodesic submanifolds, in one of a certain class of locally symmetric spaces, and certain automorphic forms, defind by theta-series, on a related symmetric space. Previous papers of the authors on the same subject are [Math. Ann. 258, 289--318 (1982; Zbl 0466.58004)] and [Invent. Math. 71, 467--499 (1983; Zbl 0506.10024)]. The original example of this phenomenon was exposed in [\textit{F. Hirzebruch} and \textit{D. Zagier}, Invent. Math. 36, 57--113 (1976; Zbl 0332.14009)], and besides the work of the authors, it has been most notably studied by \textit{S. S. Kudla} [Invent. Math. 47, 189--208 (1978; Zbl 0399.10030)], and by \textit{S. S. Kudla} and \textit{J. J. Millson} [Compos. Math. 45, 207--271 (1982; Zbl 0495.10016)]. The phenomenon is as follows. Consider the indefinite unitary group \(U(p,q)=G\). Its symmetric space \(X=U(p,q)/(U(p)\times U(q))\) can be identified with the set of positive \(p\)-dimensional subspaces of \(\mathbb C^{p+q}\) equipped with the standard Hermitian form of signature \((p,q)\). Let \(U\subseteq\mathbb C^{p+q}\) be a positive \(r\)-dimensional space, and let \(X_ U\subseteq X\) be the subset of \(X\) consisting of \(p\)-planes which contain \(U\). Then \(X_ U\) is a totally geodesic submanifold of \(X\) of codimension \(rq\). If \(G_ U\subseteq G\) is the stabilizer of \(V\), then \(G_ U\simeq U(r)\times U(p-r,q)\), and \(X_ U\) is identifiable to the symmetric space of \(G_ U\). Suppose \(\Gamma\subseteq G\) is a torsion-free discrete group. Assume for simplicity that \(U(p,q)/\Gamma\) is compact. If \(G_ U/(\Gamma \cup G_ U)\) is also compact then \(X_ U\) projects to a smooth submanifold \(C_ U\) in the quotient space \(X/\Gamma\), and so defines a codimension \(rq\) cycle in \(X/\Gamma\). These cycles \(C_ U\) are the cycles involved in the story. Now let \(k_ 1\) be a totally real number…
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 58A14
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 32Q99
Mathematics Subject Classification ID: 32C37
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 22E55
zbMATH DE Number: 3891470
zbMATH Keywords: geodesic cycles
zbMATH Keywords: Poincaré dual form
zbMATH Keywords: families of cycles
zbMATH Keywords: totally geodesic submanifolds
zbMATH Keywords: locally symmetric spaces
zbMATH Keywords: automorphic forms
zbMATH Keywords: theta-series
zbMATH Keywords: cohomology class
zbMATH Keywords: Poincaré duality
zbMATH Keywords: harmonic differential form
zbMATH Keywords: Hermitian form
reviewed by: Roger Howe
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214509287
OpenAlex ID: W1483139885
Wikidata QID: Q115173504
DOI: 10.4310/JDG/1214509287
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description: scientific article; zbMATH DE number 3891470