Differential of a period mapping at a singularity
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 image of \(\mathcal{M}\) by a canonically extended period mapping? How can Hodge theory be used to help understand the geometry of the stratification of \(\mathcal{M}\)? For the case of varieties whose period domain is Hermitian symmetric (algebraic curves, abelian varieties, K3's, hyperKähler varieties, cubic threefolds and fourfolds, \dots) this question is classical and is the subject of ongoing work. There is evidence that at least in particular cases this general question may have an interesting and occasionally surprising answer, [\textit{M. Franciosi} et al., Math. Nachr. 290, No. 5--6, 794--814 (2017; Zbl 1388.14104); \textit{P. Griffiths}, Contemp. Math. 766, 163--200 (2021; Zbl 1464.14014); ``Some geometric applications of Hodge theory'', Talks in Chile and Montreal (2020), \url{https.//hdl.handle.net/20.500.12111/7923}]. Another type of motivating question is to establish Torelli results on boundary components of a moduli space, and from these deduce Torelli results in the interior. The point here is that although they are singular the varieties that correspond to boundary points may be easier to prove Torelli type results about, and then these can be used to infer similar results in the interior, [\textit{R. Friedman}, Ann. Math. Stud. 106, 183--208 (1984; Zbl 0578.14007); \textit{G. Pearlstein} and \textit{Z. Zhang}, Algebr. Geom. 6, No. 2, 132--147 (2019; Zbl 1427.14071)]. The geometric case is when we start with a family \(X^* \to \Delta^*\) of smooth varieties whose associated period mapping \(\Delta \to \Gamma \setminus D\) has monodromy \(T = T_sT_u\). By semi-stable reduction, or by some other construction, we can obtain a family \(\pi:X \to \Delta\) where the \(X_t = \pi^{-1}(t)\) are smooth for \(t \ne 0, X_0\) is a reduced normal crossing divisor and the monodromy is unipotent. We can and will assume that the finite group \(\{T^k_s : k \in \mathbb{Z}\}\) acts equivariantly on \(X \to \Delta\) and that the induced action on \(\Delta\) has the origin as an isolated fixed point. We set \(X = X_0\) and shall use the standard identification \[ T\mathrm{Def}(X) = \mathbb{E}xt^1_{\mathcal{O}_X} (\Omega^1_X, \mathcal{O}_X) \] The local to global spectral sequence of Ext's gives \[ \begin{tikzcd} H^1(\mathcal{E}xt^0_{\mathcal{O}_X} (\Omega^1_X, \mathcal{O}_X)) \ar[r] \ar[d, equal] &\mathbb{E}xt^1_{\mathcal{O}_X} (\Omega^1_X, {\mathcal{O}_X}) \ar[r] \ar[d, equal] &H^0(\mathcal{E}xt^1_{\mathcal{O}_X} (\Omega^1_X, {\mathcal{O}_X})) \ar[d, equal]\\ T \text{Def}^{es}(X) \ar[r] &T \text{Def}(X) \ar[r] &T \text{Def}^{sm}(X) \end{tikzcd} \] where the left-hand term is the first order equisingular deformations of \(X\) and where the right-hand term (a quotient space) are the equivalence classes of partially smoothing deformations. The analysis of \(\delta\Phi\) can roughly be organized into the following cases: \begin{itemize} \item \(N = 0\): The either \(X_0 = X\) is smooth, \(\Phi(0) = H^n(X)\) lies in a Mumford-Tate subdomain \(D' \subset D\), and the condition \(\delta \Phi= 0\) means that \(\Phi(X_t)\) leaves \(D'\). Or \(X_0\) is singular and \(\delta\Phi\) measures either the variation of the Hodge structure in the equisingular directions or the Hodge theoretic properties of the smoothing of \(X_0\). \item \(N \ne 0\): Then the family will either be equisingular or smoothing and the successive terms in the associated graded to the weight filtration of \(\delta\Phi\) describe Hodge theoretic properties of either the variation of the mixed Hodge structure in the \(\ker (N)\) part of the limiting mixed Hodge structure, or of the smoothing itself. \end{itemize} This paper studies properties of the first order variation of the Hodge structure at \(t = 0\), both when the monodromy is finite, and for the case when the monodromy is infinite. In the latter case, the authors give a Torelli type result that may be used to infer Torelli properties in the interior of a moduli space from these properties on the boundary.
scientific article; zbMATH DE number 7608367