Moduli of framed manifolds
type: Item
instance of: scholarly article
title: Moduli of framed manifolds
zbMATH Open document ID: 0901.32016
author: Georg Schumacher
published in: Inventiones Mathematicae
publication date: 1998-08-10T00:00:00Z
review text: Given a homogeneously polarized manifold \(X\) the polarization can be represented by a smooth divisor \(C\) such that \(K_X+C\) is ample by \textit{T. Fujita}'s theorem [`Classification theories of polarized varieties', Cambridge University Press, New York (1990; Zbl 0743.14004)]. The framed manifold \((X,C)\) is called canonically polarized. The class of non-uniruled framed manifolds is introduced (including canonically polarized manifolds). A deformation theory is based on \textit{V. P. Palamodov}'s tangent cohomology [Russ. Math. Surveys 31, No. 3, 129-197 (1976); translation from Usp. Mat. Nauk 31, No. 3(189), 129-194 (1976; Zbl 0347.32009)]. The existence of a coarse moduli space of non-uniruled polarized framed manifolds is shown. The complement \(X''= X \backslash C\) is provided with a unique complete Kähler-Einstein metric of constant Ricci curvature \(-1\) according to theorems of \textit{G. Tian} and \textit{S. T. Yau} [Adv. Ser. Math. Phys. 1, 543-559 (1987; Zbl 0687.14032)] and \textit{R. Kobayashi} [Osaka J. Math. 21, 399-418 (1984; Zbl 0582.32011)]. The variation of complete Kähler-Einstein metrics in a holomorphic family gives rise to a generalized Petersson-Weil metric based on \(L^2\)-cohomology in the sense of \textit{St. Zucker} [Ann. Math., II. Ser. 109, 415-476 (1979; Zbl 0446.14002)]. The corresponding Kähler form on the moduli space satisfies a fiber integral formula, and the curvature tensor of the generalized Petersson-Weil form is computed. For moduli of smooth hypersurfaces in a fixed manifold \(X\), divided by \(\text{ Aut}(X)\), the holomorphic sectional curvature is bounded from above by a negative constant, yielding hyperbolicity.
reviewed by: Georg Schumacher
Mathematics Subject Classification ID: 32G13
Mathematics Subject Classification ID: 32Q15
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 1186382
zbMATH Keywords: moduli of polarized manifolds
zbMATH Keywords: Petersson-Weil geometry
zbMATH Keywords: curvature
zbMATH Keywords: hyperbolicity
MaRDI profile type: Publication
full work available at URL: s002220050264
OpenAlex ID: W2032028674
DOI: 10.1007/S002220050264
description: scientific article; zbMATH DE number 1186382