Classifying Spaces of Degenerating Polarized Hodge Structures. (AM-169)

Kazuya Kato
Sampei Usui
Pages: 352
https://www.jstor.org/stable/j.ctt6wpzkw

1. Front Matter
(pp. i-vi)
(pp. vii-xii)
3. Introduction
(pp. 1-6)

This book is the full detailed version of the paper [KU1].

In [G1], Griffiths defined and studied the classifying spaceDof polarized Hodge structures of fixed weightwand fixed Hodge numbers (hp,q). In [G5], Griffiths presented a dream of adding points at infinity toD. This book is an attempt to realize his dream.

In the special case$w=1,\quad {{h}^{1,0}}={{h}^{0,1}}=g,\quad \text{other }{{h}^{p,q}}=0,\caption{(1)}$the classifying spaceDcoincides with Siegel’s upper half space${{\mathfrak{h}}_{g}}$of degreeg. If${g=1, {\mathfrak{h}}_{g}}$is the Poincaré upper half plane$\mathfrak{h}=\left\{ x+iy\, |\, x,y\in \text{\bf {R}},y>0 \right\}$. For a congruence subgroup Γ of SL(2,Z), that is, for a subgroup of SL(2,Z) which...

4. Chapter Zero Overview
(pp. 7-69)

In this chapter, we introduce the main ideas and results of this book.

In Section 0.1, we review the basic idea of Hodge theory. In Section 0.2, we introduce the basic idea of logarithmic Hodge theory. In Section 0.3, we review classifying spacesDof Griffiths (i.e., Griffiths domains) as the moduli spaces of polarized Hodge structures. In Section 0.4, we describe our toroidal partial compactifications of the classifying spaces of Griffiths and our result that they are the fine moduli spaces of polarized logarithmic Hodge structures. In Section 0.5, we describe the other seven enlargements ofDin the...

5. Chapter One Spaces of Nilpotent Orbits and Spaces of Nilpotent i-Orbits
(pp. 70-74)

We recall polarized Hodge structures in Section 1.1, and the classifying spaceDof polarized Hodge structures in Section 1.2 (cf. [G1]). In Section 1.3, we introduce the space of nilpotent orbits${{D}_{\Sigma }}$and the space of nilpotenti-orbits$D_{\Sigma }^{\sharp}$in the directions in Σ as enlargements ofD. The space${{D}_{\Sigma }}$is the main object in this book.

Letwand$({{h}^{p,q}})={{({{h}^{p,q}})}_{p,q\in \text {\bf Z}}}$be as in Section 0.7.

AHodge structureof weightwand of Hodge type (hp,q) is a pair (HZ, F) consisting of a free Z-moduleHZof$\sum\nolimits_{p,q}{{{h}^{p,q}}}$and of a decreasing filtrationFonHC...

6. Chapter Two Logarithmic Hodge Structures
(pp. 75-106)

In Section 2.1, we recall basic facts about logarithmic structures (cf. [Kk1]). In Section 2.2, we recall the ringed spaces$({{X}^{\log }}, {\cal O}_{X}^{\log })$introduced in [KkNc], with some generalizations. We study properties of local systems onXlogin Section 2.3. In Section 2.4, we introduce the notion of the polarized logarithmic Hodge structure, which is defined on the ringed space$({{X}^{\log }}, {\cal O}_{X}^{\log })$. In Section 2.5, we observe the relationship of polarized logarithmic Hodge structures and nilpotent orbits. In particular, we interpret the nilpotent orbit theorem of Schmid in the language of polarized logarithmic Hodge structures. Further, we introduce the notion of the...

7. Chapter Three Strong Topology and Logarithmic Manifolds
(pp. 107-145)

In this chapter, we define a structure of a logarithmic local ringed space over C on$\Gamma \backslash {{D}_{\Sigma }}$(see Section 3.4), where Σ is a fan in${{\mathfrak{g}}_{\text{\bf Q}}}$and Г a subgroup ofGZthat is strongly compatible with Σ. For this, in Section 3.3, we define for each σ ∈ Σ a subsetEσof an fs logarithmic analytic spaceĚσ(a logarithmic version of the subsetDofĎ) and a map${{E}_{\sigma }}\to \Gamma \backslash {{D}_{\Sigma }}$whose image covers$\Gamma \backslash {{D}_{\Sigma }}$when σ ranges in Σ. We endowEσwith the so-called “strong topology” introduced in Section 3.1, and with the inverse images...

8. Chapter Four Main Results
(pp. 146-156)

In this chapter, we state the main results of this book: Theorem A in Section 4.1, and Theorem B in Section 4.2. We extend Griffiths’ period maps in Section 4.3 and the infinitesimal period maps in Section 4.4.

We state our first main theorem, whose proof will be given in Chapter 7 below. A subgroup Γ ofGZis said to beneatif, for eachγ∈ Γ, the subgroup of C×generated by all the eigenvalues ofγis torsion-free. It is known that there exists a neat subgroup ofGZof finite index.

Theorem ALetΣ...

9. Chapter Five Fundamental Diagram
(pp. 157-174)

The aim of this chapter is to construct the following “fundamental diagram” which gives an illustration of various enlargements ofDand of our method to prove Theorems A and B:$\begin{array}{ccccccccc} & & & & {{D}_{\text{SL}(2),\text{val}}} & \hookrightarrow & {{D}_{\text{BS,val}}} & & \\ & & & & \downarrow & & \downarrow & & \\ & & D_{\Sigma ,\text{val}}^{\sharp} & \to & {{D}_{\text{SL(2)}}} & {} & {{D}_{\text{BS}}} & \to & {{\chi }_{\text{BS}}} \\ & & \downarrow & & & & & & \\ \Gamma \backslash {{D}_{\Sigma }} & \leftarrow & D_{\Sigma }^{\sharp} & & & & & & \\ \end{array}\caption {(5.0.1)}$

In this diagram,${{D}_{\Sigma }}$,$D_{\Sigma }^{\sharp}$,$D_{\Sigma ,\text{val}}^{\sharp}$,DSL(2),DSL(2),val,DBS,val, andDBSare enlargements ofD. The first two appeared in Section 1.3, the last four were defined in our previous paper [KU2], as reviewed in Sections 5.1 and 5.2 below, and$D_{\Sigma ,\text{val}}^{\sharp}$will be defined in Section 5.3. Here, χ denotes the space of all maximal compact subgroups ofGR and χBSdenotes the Borel-Serre...

10. Chapter Six The Map $\psi :D_{\text{val}}^{\sharp}\to {{D}_{\text{SL}(2)}}$
(pp. 175-204)

This chapter is devoted to the proofs of Theorems 5.4.2, 5.4.3, and 5.4.4. In Section 6.1, we recall the theory of SL(2)-orbits in several variables in [CKS]. We prove Theorem 5.4.2 in Section 6.2, Theorem 5.4.3 (i) in Section 6.3, and Theorem 5.4.3 (ii) and Theorem 5.4.4 in Section 6.4.

In this section, we review some results in [CKS]. We review in 6.1.1 the theory of the associated SL(2)-orbit in one variable, in 6.1.2 the theory of R-split mixed Hodge stuructures, and in 6.1.3 the theory of the associated SL(2)-orbit in several variables. In the remaining part of this section,...

11. Chapter Seven Proof of Theorem A
(pp. 205-225)

In this chapter we prove Theorem A. In Section 7.1, we prove Theorem A (i). In Section 7.2, we study the actions of σC(resp.iσR) onEσ,Eσ,val(resp.$E_{\sigma }^{\sharp}$,$E_{\sigma ,\text{val}}^{\sharp}$), and then, using these results, we prove Theorem A for$\Gamma {{(\sigma )}^{\text{gp}}}\backslash {{D}_{\sigma }}$in Section 7.3. In Section 7.4, we complete the proof of Theorem A for$\Gamma \backslash {{D}_{\Sigma }}$.

In Chapter 7, let Σ be a fan in${{\mathfrak{g}}_{\text{\bf Q}}}$and let Γ be a subgroup ofGZwhich is strongly compatible with Σ.

In this section, we prove Theorem A (i).

We first prove

Proposition 7.1.1Letσ...

12. Chapter Eight Proof of Theorem B
(pp. 226-243)

In this chapter, we prove Theorem B stated in Section 4.2. In the proof of Theorem B, we use the results in Theorem A which are stated in Section 4.1 and proved in Chapter 7.

In Section 8.4, we prove Theorem 0.5.29.

We will study moduli of pairs (X, a polarized Hodge structure onX) comparing it with moduli of pairs (X, a local system onXlog). The latter is something like to forget the Hodge filtrationFin the former. In this section, we give preparations on moduli of local systems onXlog.

LetXbe an object of...

13. Chapter Nine ♭-Spaces
(pp. 244-250)

In this chapter, we will consider the relationship between our theory and the theory of Cattani and Kaplan [CK1] and discuss related subjects.

Definition 9.1.1

(i)We define$\begin{array}{l} \chi _{\text{BS}}^{\flat }:={{\chi }_{\text{BS}}}_{/}\sim ,\quad D_{\text{BS}}^{\flat }:={{D}_{\text{BS}}}_{/}\sim,\quad D_{\text{BS,val}}^{\flat }:={{D}_{\text{BS,val/}}}\sim, \\ where\text{ }x\sim y\Longleftrightarrow \left\{ \begin{array}{l} the\text{ }parabolic\text{ }subgroups\text{ }associated\text{ }with\text{ }x \\ and\text{ }y\text{ }coincide,\text{ }say\text{ }P,\text{ }and\text{ }y\text{ }\in \text{ }{{P}_{u}}x\text{.} \\ \end{array} \right. \\ \end{array}$

(ii)We define$\begin{array}{l} D_{\text{SL}(2)}^{\flat}:={{D}_{\text{SL}(2)/}}\sim ,\quad where\text{, }for\text{ }x\in {{D}_{\text{SL}(2),m}},\ y\in {{D}_{\text{SL}(2),n}}, \\ x\sim y\Longleftrightarrow \left\{ \begin{array}{l} m\text{ }=\text{ }n,\text{ }the\text{ }family\text{ }of\text{ }weight\text{ }filtrations\text{ }of\text{ }x\text{ }and \\ that\text{ }of\text{ }y\text{ }coinside,\text{ }say\text{ }W,\text{ }and\text{ }y\in {{G}_{W,\text{\bf R},u}}x.\\ \end{array} \right.\\ \end{array}$

The space$\chi _{\text{BS}}^{\flat}$was studied by Zucker in [Z1], [Z4]. This space is called the “reductive Borel-Serre space” by him.

In [KU2, 3.15, 3.16], we announced that we will discuss in this book extended period maps into$D_{\text{SL}(2)}^{\flat}$. However, we later realized that$\Gamma \backslash D_{\text{SL}(2)}^{\flat}$for a subgroup Γ ofGZof finite index is not Hausdorff in general and now we are not sure whether this...

14. Chapter Ten Local Structures of DSL(2) and $\Gamma \backslash \text{D}_{\text{SL}(2),\le 1}^{\flat }$
(pp. 251-270)

In Section 10.1, for eachp= [ρ,φ] ∈DSL(2)of rankn, we give a homeomorphism from an open neighborhood ofpto a certain subspace of$({\bf R}_{\ge 0}^{n}) \times {{\mathfrak{g}}_{\text{\bf R}}}\times ({{K}_{\text{r}}} \cdot \bf {r)}$, wherer=φ(i) and${{K}_{\text{\bf r}}}\cdot \text{\bf r}=\{k\cdot \text{\bf r}|k\in {{K}_{\text{\bf r}}}\}\subset D$, which sendspto (0, 0, r) (Theorem 10.1.3). ThoughDSL(2)is not a real analytic manifold, this homeomorphism is something like a “real analytic local coordinate” ofDSL(2). From this theorem, we can obtain Theorem 10.1.6 which is a criterion for the local compactness ofDSL(2). We give the proof of Theorem 10.1.3 in Section 10.3 after we consider a...

15. Chapter Eleven Moduli of PLH with Coefficients
(pp. 271-276)

In this chapter, we generalize Theorems A and B to the case of polarized logarithmic Hodge structures with coefficients. The key observation is Lemma 11.1.3. Then the generalized results Theorems 11.1.17 and 11.3.1 follow from Theorems A and B, respectively.

In §11, letAbe a semi-simple Q-subalgebra of EndQ(H0,Q) such that, for anyaA, the dual mapping$^{t}a:{{H}_{0,\text{\bf Q}}}\to {{H}_{0,\text{\bf Q}}}$with respect to${{\left\langle \, , \right\rangle }_{0}}$belongs toA. Heretais characterized by${{\left\langle ax,y \right\rangle }_{0}}={{\left\langle x{{,}^{t}}ay \right\rangle }_{0}}$for anyx,yH0,Q.

LetĎAĎbe the subset ofĎconsisting of allFĎsuch that theFpare...

16. Chapter Twelve Examples and Problems
(pp. 277-306)

In this chapter, we give some examples and open problems.

In this section, we consider Siegel upper half spaces. We describe some points at infinity of Siegel upper half spaces, and consider their behaviors in the map$\psi :D_{\Sigma ,\text{val}}^{\sharp}\to {{D}_{\text{SL(2)}}}$(Section 5.4, Chapter 6).

In Section 12.1, we assumew= 1,h1,0=h0,1=g,hp,q= 0 if (p,q) ≠ (1, 0), (0, 1), and there is a Z-basis${{({{e}_{j}})}_{1\le j\le 2g}}$ofH0such that${{\left( {{\left\langle {{e}_{j}},{{e}_{k}} \right\rangle }_{0}} \right)}_{j,k}}=\left( \begin{matrix} 0 & -{{1}_{g}} \\ {{1}_{g}} & 0 \\ \end{matrix} \right).$

By using this basis, we identifyH0,R=R2gforR= Z, Q, R, C. The algebraic groupGis identified with...

17. Appendix
(pp. 307-314)
18. References
(pp. 315-320)
19. List of Symbols
(pp. 321-330)
20. Index
(pp. 331-336)