LOG HOD，GE THEORY AND AN

APPLICATION TO CALABI-YAU THREEFOLDS

SAMPEI USUI

ABSTRACT. This is a talk about log Hodge theory constructed in the book:

K. Kato and S. Usui， Clαssザyingspαces of degenerating po如何zedH odge structures， Ann. of Math. Stud.， 169， Princeton Univ. Press， Princeton， NJ， 2009.

As an application， a generic Torelli theorem of quintic-mirror is proved in section 2. The next short poem is at the top of this book.

Suugαkuωα mugen enten kou kokoro Koute kogαrete harukana tαbiji

by Kazuya Kato and Sampei Usui， which was translated by Luc Illusie as

L 'impossible voyαgeαux points a l 'infini N'αpαs fait bαttre en vαin le coeur du geometre

CONTENTS

gO. Introduction g1. Log Hodge theory (with K. Kato) g2. Quintic-mirror

90. INTRODUCTION

This part is a summary of the introduction of the book with K. Kato.

Gri:ffiths defined and studied i:o. [G 1] the classifying space D of polarized Hodge struetures of fi.xed weightωand :fi.xed Hodge numbers (hp，q)， and presented in [G2] a dream to add points at infinity to D. [KU1] is an announcement of our attempt to realize his dream， and this book is its full-detailed version.

In a special case w here ω= 1， h1，o = hO，l = g， and other hp，q = 0， the classiちTingspace D coincides with Siegel's upper half space ~g of degree g. In this case， for a subgroup r of Sp(g， Z) of finite index， toroidal compacti五cationsof r\~g ([AMRT]) and the Satake-Baily-Borel compacti五c抗ionof r\~g ([Sa]， [BB]) are already constructed， where points at泊五nityoften play more important roles than usual points. For example，

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in the simples七case9 = 1， the Taylor expansion of a modular form at the standard cusp (i.e・ぅ the class of∞ ε Pl(Q) rωdulo r) ofthe compactified modular curve r¥(f)UPl(Q) ) is called the q-expansion and is very important in the theory of modular forms.

The theory of七hesecompactifications is included in a general theory of compactifica-tions of quotients of symmetric Hermitian domains by the actions of discre七earithmetic groups. Howeverぅ theclassifying space D in general is rarely a symmetric Hermitian domain， and we can not use the general theory of symmetric Hermi七iandomains when we七ryto add points-at infi凶七Yto D. In this book， we overcome七hisdiffic叫ty.We discuss two subjec七s.

Subject I. Toroidal partial compactifications and moduli of polarized logarithmic Hodge structures.

In this book， for general D， we construct a kind of toroidal partial compactification r¥DE ofr¥D associated to a fan E and a discrete subgroup r of Aut(D) satisむTinga certain compatibility with E.

In the case D = f)gぅtheclasses of polarized Hodge structures in r¥f)g converge to a point at infinity of r¥f)g when the polar包edHodge structures degenerate. As in [Sc]， nilpotent orbits appear when polarized Hodge structures degenerate. In our definition of DE for general D， a nilpotent orbit i七selfis viewed as a point at infinity. In order七odo so， we use logari七hmics七ructuresintroduced by Fontaine and Illusie and developed in [K]， [KN]. The theory of nilpotent orbits is regarded as a local aspect of the theory of polarized logarithmic Hodge structures (= PLH). A fundamental observation here is: (a nilpotent orbit) = (a PLH over a loga比 hmicpoin七).

Our main theorem concer凶ngSubject 1 is stated roughly as follows.

Theorem. r¥DE is the fine moduli spαce 01 '争olαrizedlogαrithmic H odge structures" withα“r -level structure" whose“local monodromiesαγe in the directions in E".

In the classical case D = f)g， for a subgroup r of Sp(g， Z) of fini七eindex and for a sufficiently big fan E， r¥DE is a toroidal compactification of r¥f)g・Alreadyin this classical case，七histheorem gives moduli-theoretic interpretations of the toroidal com-

pactifications of r¥f)g.

For general D， the space r¥DE has a kind of complex structure， but a delicate point is that七hisspace can have locally the shape of “complex analytic space with a slit" (for exampleぅc2minus {(O， z) I z εCぅzチO})， and hence it is often not locally compact. However i七isvery near七oa complex analy七icmanifold， and we call it a “logarithmic ma凶fold".Infi凶 tesimalcalculus is performed on r¥DE nicely. These phenomena are firs七examinedin the easiest non-七rivial case in [U1].

One motivation of Griffi七hsfor adding poin七sat infinity to D was the hope that the period mapム*→ r¥D，associa七edto a variation of polarized Hodge structure on a punctured discムヘ couldbe ex七endedover the puncture. By using the above main theorem and七henilpotent orbi七theoremof Schmid， we can actually extend七heperiod

map toム→ r¥DEfor some suitable fan E.

Subject II. The eight enlargemen七sof D and the fundamental diagram.

In七heclassical case D =均的七hereis aωther compactification r¥DBS of r¥f)g， for r a subgroup of Sp(g， Z) of finite index， called Borel-Serre compactification ([BS]). This is a real manifold with corners (like R雪。 x R九 locally).

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For general D， by adding七oD points at infi.凶tyof different kinds， we obtain eight enlargements of D with maps among them whlch form the following diagram.

F'undamenta1 Diagram.

DSL(2)，val L→ DBs，val

Dr;.val 十一 DZO ，val ー→ DSL(2) DBS ，

Dr; +- D~

Note七hatthe space Dr;， which appeared in Subject 1， si七sat the left lower end of this diagram. Like nilpo七entorbits， SL(2)ーorbitsalso appear in七heぬeoryof degenera七ionsof polarized Hodge structures ([Sc] ， [CKS]). This diagram悦11show nilpotent orbits， SL(2)-orbits， and the theory of Borel-Serre are related. The lef七回handside of the above diagram has Hodge-出eoreticn抗ure，and the right四 handside has七henature of theory

of a恥 braicgroups. These are related by七hemiddle map'D~，val → D乱(2) which is a geometric interpretation 0ぱfSL(伊仰2幻)四Oぽrbi抗色 七heorem0ぱfCa抗t七切an凶i司 Ka叩.plan任.-Scぬhm

The eight Sp配 esDr;， D~ ， D乱(吟 DBS，Dr;，vab D~，val' D乱 (2)，vabDBS，val in Funda-mental Diagram are defined as the spaces of nilpo七entorbits， nilpotent i-orbits， SL(2)-orbits， Borel-Serre orbits， valuative nilpotent orbits， valua七ivenilpoten七i-orbits，valua-tive SL(2)ーorbits，valuative Borel-Serre orbits， -respectively. Roughly speaking， r¥Di:， is

like an analytic manifold with slits， D~ and D乱 (2)a削 ikereal manifolds with corners

and slits， DBS is a real manifold with cornersぅr¥Di:"valand D~，val are the projective limi七sof "blowing-ups" of r¥Dr; and D~ ， respectively， associated to rational subdivi-sions of 2::， DSL(2)，val and DBS，val are the projective limits of cert~in "blowing-ups" of

DSL(2) and DBS， respectively. The maps r\D~ → r\Dí:， and r¥DL，val→r¥Ds，val are proper surjective maps， described by logarithmic struc七ures，whose五bersare products

of白utecopies of S 1 . In the classical case D == ~g ， we have DSL(2) == DBS and DSL(2)，val == DBS，val in

Fundamental Diagram. This is because， for such simple polar包esHodge struc七uresぅ

all filters appeared in七heassocia七edset of monodoromy-weigh七五ltrationsare totally lineaI匂 orderedby inclusion and七heGriffiths transversality becomes a vacant condi-tion. Fundamental Diagram gives a relation between toroidal compac七ificationsr¥Di:，

ofr\~ 9 and the Borel-Serre compacti五cationr¥DBS of r\~g ・ The Satake-Baily-Borel compactification sits under r¥DSL(2) == r¥DBS・ Alreadyin this classical case，七hese

relations were not known before.

In this book， we study all these eight spaces. To prove七hemain theorem in Subject 1 and to prove that r¥Dr; has good properties such as Hausdorff property，凶cein五ni七esi-mal calculus弓 etc.，we need七oconsider all the eight spaces; we discuss from the right to theleft in the R1ildamental Diagram to deduce nice properties ofr¥Dr;， starting from

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Added this time. [U2] ， [U3] are some geometric applica七ionsof the present res叫 s.A new project [KNU]， which is組 evolutionfor log，出凶icmixed Hodge struct旧民 isln progress.

Acknowledgment. The speaker would like to express his gratitude to the organizers of 8th Oka Symposium.

[AMRT]

[BB]

[BS]

[CKS]

[G1]

[G2]

[K]

[KN]

[KNU]

[KU1]

[KU2]

[Sa]

[Sc]

[U1]

[U2] [U3]

REFERENCES

A. Ash， D. Mumford， M. Rapoport and Y. S. Tai， Smooth compactポcαtionof locαlly sym-metric variet犯s，Math. Sci. Press， BrookIine， 1975. L. Baily and A. Borel， Gompαct併ωtionofαばthmeticquotients of bounded symmetric do-mains， Ann. of Math. 84 (1966)， 442-528. A. Borel and J .-P. Serre， Gorners αnd arithmetic groups， Comment. M叫 h.Helv. 48 (1973)， 436-491. E. Catta凶， A. Kaplan and W. Schmid， Degeneration of Hodge structures， Ann. of Math. 123 (1986)， 457-535. P. A. Griffiths， Periods of仇tegralson algebraic manifolds. 1. Gonstructionαnd properties of modulα，r variet記s，Amer. J. M抗 h.90 (1968)， 568-626. 一一一ーゥ Periodsof integrals onα1gebraic mαnifolds: Summary of main results and discus-sions of open problems， Bull. Amer. M叫 h.Soc. 76 (1970)， 228-296. K.K抗 0，Logα，rithmic structures of Fontaine-Illusie， in“Algebraic組 乱lysis，geometry， and number theory" (J.-1. Igusa， edふJohnsHopkins Unive凶 tyPress， Baltimore， 1989， pp. 191-224. K. Kato and C. Nakayama， Log Betti cohomology， log et，αle cohomology， and log de Rhαm cohomology of log schemes ωerC， Kodai M抗 h.J. 22 (1999)， 161-186‘

K. Kato， C. N叫cayamaand S. Us瓜 Glωsifyingspaces of d句 eneratingmixed Hodge struc-tures" Part 1 (appe町 ed)，Part II (in preparation)， Part III (in prep紅抗ion)，. K. Kato and S. Usui， Logαrithmic Hodge structures and classifying spαces， CRM Proc. & Lect. Notes : The Arithmetic and Geometry of AIgebraic Cycles， (NATO Advanced Study Institute / CRM Summer School 1998: Ban在， C乱nada)24 (1999)， 115-130. 一一一一，Borel-Serナ'espaces and spαces of SL(2)-orbiお， Adva即 edStudies in Pure Math. 36 Algebraお Geometry2000， Azumino， (2002)，321-382. 1. Satake， On compαctificαtions of the quotient spaces forαrithmetically defined disωnt仇-

uous groups， Ann. of M抗 h.72 (1960)， 555-580. W. Schmid， Variation of Hodge structure: the singularities of the period mαpping，Invent. Math. 22 (1973)， 211-319. S. Usui， Gom，ηLψpμleωx str問t包ωL必ct'l.町L町，re“s0仰ηp仰α7悩t制4ぬαalω mp仰αct~析ficωαti抑4旬0'1ηts 0ザfαTバit仇hmet'説枕4化cquot託t詑的e臼?η1お oザIfcl αωss叫i狗fかy-仇.gs叩pαωce白soザtfHo吋dgμest門問Lcture一一一，Imαgesof extended period maps， J. AIg. Geom. 15..4 (2006)， 603-621. 一一一_， Generic To何 llitheorem for mi作 or-qunticfamily， Proc. J apan Acad.， Ser. A， 84A (8) (2008)， 143-146.

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