On the Shimura variety having Mumford's fake projective plane as a connected component
type: Item
instance of: scholarly article
title: On the Shimura variety having Mumford's fake projective plane as a connected component
zbMATH Open document ID: 1140.14020
author: Fumiharu Kato
published in: Mathematische Zeitschrift
publication date: 2008-05-27T00:00:00Z
review text: A fake projective plane (FPP) is a compact complex surface with the numerical invariants \(c_1^2=3c_2=9, p_g=q=0\) and with an ample canonical class \(K_X\). FPP has the same Betti numbers as \(\mathbb{C}\mathbb{P}^2\) but not homeomorphic to it. The first example of FPP was constructed by \textit{D. Mumford} [Am. J. Math. 101, 233--244 (1979; Zbl 0433.14021)]. Any FPP can be realized via \(p\)-adic uniformization as an arithmetic quotient of the complex unit ball. The main purpose of this paper is to give an explicit description of the Shimura variety that has Mumford's FPP as a connected component. A Shimura variety is constructed as a disjoint union of finitely many arithmetic quotients of a complex unit ball. The reflex field associated to it is \(K\) is the imaginary quadratic field \(\mathbb{Q}(\sqrt{-7})\). Theorem. The Shimura variety consists of three connected components, and has the canonical model defined over the reflex field \(K\). Furthermore, these connected components are all isomorphic to each other as complex surfaces, and one of which is isomorphic to Mumford's FPP. These connected components have models defined over the \(7\)-th cyclotomic field \(\mathbb{Q}(\zeta_7)\) over \(\mathbb{Q}\).
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 14J29
Mathematics Subject Classification ID: 11G15
zbMATH DE Number: 5279025
zbMATH Keywords: Mumford's fake projective plane
zbMATH Keywords: Shimura variety
zbMATH Keywords: unitary group
zbMATH Keywords: p-adic uniformization
reviewed by: Noriko Yui
MaRDI profile type: Publication
full work available at URL: s00209-007-0241-1
OpenAlex ID: W2074030709
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DOI: 10.1007/S00209-007-0241-1
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description: scientific article; zbMATH DE number 5279025