The moduli space of Enriques surfaces and the fake monster Lie superalgebra
type: Item
instance of: scholarly article
title: The moduli space of Enriques surfaces and the fake monster Lie superalgebra
zbMATH Open document ID: 0886.14015
DOI: 10.1016/0040-9383(95)00036-4
author: Richard E. Borcherds
published in: Topology
publication date: 1998-04-23T00:00:00Z
review text: It is known that the moduli space \(D^0\) of an Enriques surface is the quotient \(D\) of a 10-dimensional hermitian symmetric space \(\Omega\) by a discrete group, with a divisor \(H_d\) removed. Hence \(D^0\) is quasi-projective. The author proves that \(D^0\) is quasi-affine, more precisely it is an affine variety with an affine line removed. The author considers a function \(\Phi\) defined previously by him [see \textit{R. E. Borcherds}, Invent. Math. 109, No. 2, 405-444 (1992; Zbl 0799.17014)] and related to the fake monster Lie superalgebra. He proves that \(\Phi\) is an automorphic form of degree \(4\) over \(\Omega\) and has a simple zero along \(H_d\) and has no other zeros. The proof uses the explicit description of \(\Phi\) as either an infinite product or an infinite sum. It follows that \(\Phi\) gives a trivialization over \(D^0\) of the ample line bundle which gives the projective embedding of \(D^0\), so that \(D^0\) is quasi-affine. The precise structure of \(D^0\) is studied by using the Bailey-Borel compactification of \(D\) obtained by \textit{H. Sterk} [Math. Z. 207, No. 1, 1-36 (1991; Zbl 0736.14017)]. The paper contains some interesting speculations about possible generalizations of the form \(\Phi\) over other symmetric bounded domains, with the hope to get period spaces of some families of Calabi-Yau n-dimensional manifolds.
reviewed by: Giorgio Ottaviani
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 11F22
zbMATH DE Number: 912196
zbMATH Keywords: moduli space of Enriques surfaces
zbMATH Keywords: fake monster Lie superalgebra
MaRDI profile type: Publication
full work available at URL: 0040-9383(95)00036-4
OpenAlex ID: W2016586504
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description: scientific article; zbMATH DE number 912196