論文 / 著書情報
Article / Book Information
題目(和文)

K3曲面の自己同型群とその群論的不変量

Title(English)

Automorphism groups of K3 surfaces and their group invariants

著者(和文)

高津大樹

Author(English)

Taiki Takatsu

出典(和文)

学位:博士(理学),学位授与機関:東京工業大学,
報告番号:甲第12633号,
授与年月日:2024年3月26日,
学位の種別:課程博士,
審査員:加藤 文元,内藤 聡,下元 数馬,野坂 武史,馬 昭平,谷田川 友里,深谷 友宏

Citation(English)

Degree:Doctor (Science),
Conferring organization: Tokyo Institute of Technology,
Report number:甲第12633号,
Conferred date:2024/3/26,
Degree Type:Course doctor,Examiner:,,,,,,

学位種別(和文)

博士論文

Category(English)

Doctoral Thesis

種別(和文)

論文要旨

Type(English)

Summary

Powered by T2R2 (Tokyo Institute Research Repository)

（博士課程）Doctoral Program論 文 要 旨ＴＨＥＳＩＳ ＳＵＭＭＡＲＹ

系・コース：

数学
数学

Department of, Graduate major in

学生氏名：
Student’s Name

髙津 大樹系コース

申請学位（専攻分野）：

博士

Academic Degree Requested

Doctor of

審査員主査：
Chief Examiner

加藤（理学）
文元

要旨（英文 800 語程度）
Thesis Summary （approx.800 English Words ）This thesis consists of two parts. First, we mainly study singular K3 surfaces withdiscriminant 3, 4, and 7. Secondly, we study automorphism groups of K3 surfaces in view ofhyperbolic geometry.
In the first part of this thesis, we focus on the singular K3 surfaces with discriminant 3, 4,and 7, and aim at finding common properties among them, which should give us clue tounderstand the geometry of these surfaces more in depth. The automorphism groups of thesethree K3 surfaces have already been calculated. Naruki gave some constructions of thesingular K3 surface of discriminant 7. One of the constructions is given by the double
covering of the projective plane branched over a specific plane sextic, which is induced fromthe quotient by the involution associated with an elliptic fibration. We will discuss thisconstruction, where we will see how one can find the explicit equation of the sextic, as Naruki
did. We will see, moreover, that this method can be extended to the cases of discriminant 3
and 4 to obtain the double covering constructions and the explicit equations of thecorresponding sextics. Interestingly, as we will show, these sextics have several properties incommon, especially in their singularity types. For example, they have a singular point of typeD4, and the pencil of lines passing through this point gives rise to the elliptic fibration on theK3 surfaces. From these common properties, one can read a general recipe to construct theseK3 surfaces, and obtain a 3-dimensional period domain. Moreover, we show that the K3surfaces that appear in this period domain are characterized in terms of the branching locus
over the projective plane, the configuration of the singular fibres of a Jacobian elliptic
fibration, and also in terms of periods. Here, by the definition, three singular K3 surfaceswith discriminant 3, 4, and 7 correspond to points in the period domain, and by the result ofNikulin, a general member of the 3-dimensional moduli space admits an infinite
automorphism group.
Recently, Sienkiewicz has discussed the construction of the singular K3 surfaces withdiscriminant 4 as the double covering of the del-Pezzo quintic, and its automorphism group.
This construction is essentially the same as the construction in this paper.
In the second part of this thesis, we discuss the classification problem of automorphismgroups of K3 surfaces, which is located at the intersection of several research fields, such as
lattice theory, reflection group theory, and finite sporadic simple group theory. Among them,
the application of Borcherds’ lattice theory is known as a useful tool for computing agenerating set of automorphism groups of K3 surfaces. The problem of finding a generating
set of the automorphism groups of Kummer surfaces was first proposed by Klein. This
problem took more than 100 years to be solved by Kondo using Borcherds' method. Although
this method is very useful and, in fact, the automorphism groups of some specific K3 surfaces
have been calculated, this method is not applicable to the general case.
However, Shigeru Mukai recently proposed the conjecture on discrete group theory andautomorphism groups, from which one may hopefully find a general framework for dealing
with infinite automorphism groups of elliptic K3 surfaces. This conjecture for elliptic K3
surfaces claims that the virtual cohomological dimension of automorphism groups of elliptic
K3 surfaces equals the maximal Mordell-Weil rank for all elliptic fibrations.
Our main result is a formula that determines the virtual cohomological dimension of theautomorphism group of a K3 surface by the covering dimension of the blown-up boundary
associated with the ample cone of the K3 surface. To show this result, we focus on the naturalaction of automorphism groups on hyperbolic spaces, which come from the action on thecohomology of degree 2. Then we can consider the symplectic automorphism group of a K3

surface as a geometrically finite subgroup of Mobius transformations of a hyperbolic space.
Bestvina and Mess showed that the cohomological dimension of a torsion-free hyperbolic
group is determined by the covering dimension of its Gromov boundary. Recently, Tomohiro
Fukaya generalized the formula for relatively hyperbolic groups satisfying a certain
condition. Geometrically finite groups are typical examples of relative hyperbolic groups.
We can immediately obtain our main result by applying the result of Fukaya.
An application of this theorem is the case that the ample cones of K3 surfaces are spherepackings. Baragar constructed K3 surfaces whose ample cones are sphere packings. Then wecan compute the virtual cohomological dimension of their automorphism groups. Note thatthese K3 surfaces provide examples that affirm Mukai’s conjecture, and, to the best of ourknowledge, they are the first non-trivial examples in the sense that the verification of theconjecture does not rely on the explicit calculation of the group structure of the automorphismgroups.

備考 : 論文要旨は、和文 2000 字と英文 300 語を 1 部ずつ提出するか、もしくは英文 800 語を 1 部提出してください。
Note : Thesis Summary should be submitted in either a copy of 2000 Japanese Characters and 300 Words (Engｌish) or 1copy of 800
Words (English).
注意：論文要旨は、東工大リサーチリポジトリ（T2R2）にてインターネット公表されますので、公表可能な範囲の内容で作成してください。
Attention: Thesis Summary will be published on Tokyo Tech Research Repository Website (T2R2).