Differential of a period mapping at a singularity
type: Item
instance of: scholarly article
title: Differential of a period mapping at a singularity
zbMATH Open document ID: 1502.14026
published in: Pure and Applied Mathematics Quarterly
publication date: 2022-10-27T00:00:00Z
review text: The study of the variation of Hodge structure in a family of algebraic varieties is an important topic in algebraic geometry. Of special interest is analysis of the Hodge structure in a family of varieties \(X_t, \ t \in \Delta =\) unit disc, where \(X_t\) is smooth for \(t \ne 0\) while \(X_0\) may be singular. We shall usually restrict to the 1-parameter case of a period mapping \(\Phi : \Delta^* \to \Gamma \setminus D\) where \(\Gamma = \{T^k : k \in \mathbb{Z}\}\) is generated by the monodromy operator. In the Jordan decomposition \(T = T_sT_u\) the semi-simple factor \(T_s\) is of finite order (its characteristic polynomial is a product of cyclotomic polynomials), and the unipotent factor \(T_u = e^N\). We will separate two parts: (a) \(N = 0\), and (b) \(N \ne 0\). In the first case, \(\Gamma\setminus D\) is an analytic variety with quotient singularities by the action of a finite group and \(\Phi\) extends to a mapping of analytic varieties \[ \Phi:\Delta \to \Gamma\setminus D \] where \(\Phi(0) \in D\) is a polarized Hodge structure having an action of \(\Gamma\). In the second case, we shall assume that \(T_s = Id\). Then \(\Phi(0) \in \exp(\mathbb{C}N) \setminus \check{D}\) where \(\check{D}\) is the compact dual of \(D\); it is given by an equivalence class of limiting mixed Hodge structures. In both cases, we shall define a differential \(\delta\Phi\) that is an invariant of the suitably interpreted first non-zero term in the expansion of \(\Phi(t)\) about \(t = 0\). Let \(\mathcal{M}\) be the canonical completion of the KSBA moduli space \(\mathcal{M}\) of a class of varieties of general type [\textit{J. Kollár}, in: Handbook of moduli. Volume II. Somerville, MA: International Press; Beijing: Higher Education Press. 131--157 (2013; Zbl 1322.14006)]. How much of the stratification of \(\mathcal{M}\) by the singularity type of the corresponding varieties or by the presence of ``additional'' algebraic subvarieties is reflected in the stratification of the …
reviewed by: Mohammad Reza Rahmati
Mathematics Subject Classification ID: 14D07
Mathematics Subject Classification ID: 14C30
zbMATH DE Number: 7608367
zbMATH Keywords: variation of Hodge structure
zbMATH Keywords: period map
zbMATH Keywords: deformation space
zbMATH Keywords: moduli of Hodge structure
author: Mark Green
author: Phillip Griffiths
MaRDI profile type: Publication
full work available at URL: pamq.2022.v18.n4.a6
OpenAlex ID: W4313125429
DOI: 10.4310/PAMQ.2022.V18.N4.A6
description: scientific article; zbMATH DE number 7608367