Fumiharu
Kato
Department of Mathematics

Kumamoto University

Professor

Email: kato “at”
sci.kumamotou.ac.jp Tel:
+81(0)963423336 
Research interests
(1) Backgrounds
Although my fields of research, broadly speaking, fall into algebraic
geometry, the problems that I have been dwelling with in my past research are
often not purely algebrogeometric, and are typically associated with other
areas of mathematics, such as number theory, analytic function theory, and
analytic geometry. For this kind of problems, most of the ‘standard’ concepts and techniques
often fail to work properly; that is to say, in the context of
algebraic/arithmetic geometry, the usual notion of schemes, algebraic
varieties, and other ordinary notions of spaces are not any more powerful
enough. Asking for more wellfunctioning techniques and devices, then, one
would want to make use of ‘new geometry’, which would provide,
expectedly, a new notion of spaces that pushes the frontier of
algebraic/arithmetic geometry, embracing those problems in ‘borderline region
mathematics’. This is my basic
attitude; in other words, my general research guideline is:
l
approaching to ‘borderline region mathematics’, where algebra, geometry and analysis
all together come to interplay, by using new notions of spaces, which expand and
strengthen the hitherto known algebrogeometric spaces.
Here, as the ‘borderline region
mathematics’ in the context of
algebraic geometry, I focus on the following two topics:
l
Topics that are related with degenerations;
l
Topics that deal with infinite quotients, such as
those appear in uniformizations.
Degeneration of geometric objects is, admittedly, a fairly classical
topic in algebraic geometry; but it often exceeds the realm of classical
algebraic geometry, as it relates itself quite naturally with analytic geometry
and number theory through, for example, analysis of period maps in Hodge
theory. It is here that the other topic, uniformizations, often comes to
interplay, giving rise to, furthermore, at least in the classical situation,
the studies in the rich field of function theory and the theory of algebraic
differential equations on algebraic/arithmetic varieties. This lastmentioned
mathematical playground, in which ‘degenerations’ and ‘uniformizations’ are so deeply
interwoven, is a very interesting fertile land in mathematics, in which
algebraic geometry, analytic geometry, function theory, and even number theory
play fundamental roles. It can be also said that, historically, lots of
mathematical streams flow from this region; even algebraic geometry itself,
having stemmed from the meeting point, essentially in the works of Riemann, of
the theory of elliptic functions and projective geometry, originates from this
parts of mathematical world. One can think of this mathematical region,
therefore, as ‘fertile crescent’ in mathematics.
In order to play around in the thus described mathematical playground,
the following ‘new notions of spaces’ often give a useful and
consistent standpoint:
l
Log schemes  in log geometry;
l
Rigid spaces  in rigid geometry.
(2) Log geometry  ‘geometry of degenerations’
Log geometry is a relatively new theory of geometry, developed by J.M.
Fontaine, L. Illusie, and K. Kato in late 1980s. The most characteristic
feature of this new geometry lies in its notion of log smoothness, by which one is able
to treat nonsingular varieties and singular varieties of some sort, such as
the ones obtained from semistable degenerations of smooth varieties, within
the uniform category, pretending that they all together are ‘smooth’. This notion of the ‘new smoothness’, so to speak, provides
entirely new viewpoints in the treatment of degenerations of algebraic
varieties and various related arithmetic phenomena.
I have started my research career with log geometry, which yielded
roughly the following three pieces of works:
A) Deformation theory  Log smooth deformation theory;
B) Moduli theory  Moduli theory of log curves;
C) Log Poincaré lemma and log de Rham Theory.
A. I developed in my master thesis in Kyoto university [1] the socalled
log smooth deformation theory, which aims at a general deformation theory
framework that allows not only classical smooth deformation but also
degeneration of algebraic varieties. This theory has been cited by many
authors, applied not only to the classical algebrogeometric problems related
to global smoothing of normal crossing varieties, but, recently, also to
various problems in mathematical physics related to, for example, mirror
symmetry.
B. I applied my log smooth deformation theory to the moduli theory of
algebraic curves. This is done in my doctor thesis in Kyoto University [6].
This piece of work actually turns out to be the first systematic endeavor to
discuss moduli theory in the framework of log geometry. Notice that, due to the
notion of log smoothness, moduli spaces of ‘smooth objects’ in log geometry almost
automatically contain some appropriately selected degenerations, and thus are,
in many cases, expected to be ‘compact’ from the beginning.
This is the reason why moduli theory in log geometry should come as a natural
thing to study. In this connection, it deserves mentioning here that, in recent
years, the moduli spaces of the socalled log stable maps have become the
object of active research in connection with mirror symmetry and GromovWitten
invariants in mathematical physics.
C. It is natural to expect that, in many situations, geometry of
degenerations is associated with several analytic problems through, for
example, the theory of limiting Hodge structures. I discussed in [2] the ‘loggeneralization’ of Poincaré lemma and de Rham
theory. This piece of work has later been applied to the socalled log Hodge
theory by K. Kato and S. Usui.
(3) Rigid geometry  ‘analytic approach to arithmetic geometry’
Rigid geometry is a new
framework of geometry, introduced by J. Tate around 1960 first as an ‘analytic’ geometry over
nonarchimedean valued fields, such as padic fields. Although this
framework has initially sprouted from the attempts to ‘copy’ complex analytic
geometry to the nonarchimedean situation, and is, in this respect, perhaps the
most successful one among several such attempts at the time when it appeared,
it has many aspects that look quite different from complex analytic geometry.
For example, rigid geometry has the notion of ‘analytic reduction’, which is, so to speak,
an ‘analytic’ analogue of the usual
modpreduction. It turns out that, in fact, the presence of
these differences is highly essential, and it is basically due to this feature
of the theory that, unlike complex analytic geometry, rigid geometry can be
applied to diverse kinds of numbertheoretic problems, such as rational points
of elliptic curves.
One of the most important aspects of rigid geometry lies in that it
enables one to deal with ‘infinite’ objects and operations,
such as infinite coverings and quotients, within the context of arithmetic
geometry. This theory, the theory of the socalled nonarchimedean uniformizations, admits extremely broad
applications, giving very effective and useful technical devices in, for
example, constructing interesting arithmetic varieties, such as Tate elliptic
curves, Mumford curves and the famous Mumford’s ‘fake projective plane’, and also in
investigating arithmeticogeometric structures and modpreductions of
arithmetic varieties. Moreover, quite interestingly, the uniformization thus
described in the padic side is, when the group is
arithmetic in an appropriate sense, closely related with the usual
uniformization in the complex analytic side through its arithmetic backgrounds
(the theory of padic uniformization of Shimura
varieties).
My works in rigid geometry are roughly classified into the following two
areas:
A) Theory of nonarchimedean orbifolds and their
uniformizations;
B) Foundations of rigid geometry.
A. My interest in rigid geometry stems from
the past works on fake projective planes [4][16][19], which I started when in
the master course. As a natural extension of these works, I have been
interested in the quotients by infinite discrete groups that contain
finiteorder elements (nonarchimedean orbifold uniformizations) and have aimed
at constructing the general theory of them. In dimension one, such a general
theory has already been done, partly by the collaboration with Gunther
Cornelissen and Aristeides Kontogeorgis [7][8][9][10][11][12][13]; one finds
the general treatment of this theory in [12]. An interesting byproduct
obtained from this thread of study is the notion of padic
Schwarzian triangle groups, which are, of course, related with some of the padic Gaussian hypergeometric differential equations. Not only the groups, but one is also
interested in the related geometrical objects; when the groups in question are
finite, they should be the ‘padic analogue’ of regular platonic solids. The padic icosahedron [14] has been thus obtained. The 2adic icosahedron
was depicted in the front cover of Notices AMS 52, No.7 (2005) (see below). Finally, there
is recent progress in the theory of nonarchimedean orbifold uniformizations in
higher dimensions in the collaboration with Daniel Allcock [25].
B. After Tate introduced his rigid geometry, Raynaud gave another way of approaching to it. Since then, several other approaches have appeared. This is of course a desirable situation, for we can choose, according to what we want to do, the most appropriate one from the bunch of possible approaches to rigid geometry. But, at the same time, one wants to have more unified and consistent picture, which would explain clearly the interrelations between these approaches. Since around 2003, Kazuhiro Fujiwara in Nagoya and I have promoted the book project ‘Foundations of rigid geometry’, in which we attempted to put the whole theory of rigid geometry on more general and sound bases, and thus to provide more clean and unified pictures of the entire theory. The most difficult point of the project lies in that, in rigid geometry, one cannot rely on Grothendieck’s EGA, since schemes and formal schemes that appear in the context of rigid geometry are almost always not Noetherian. This implies that, in order to provide a sound enough foundations of rigid geometry, one has more or less to reconstruct quite a few portions of EGA, and implement more powerful and broadly applicable devices and techniques. The first volume of the book [24] will soon appear from European Mathematical Society publisher.
List of publications (list in MathSciNet)
Research papers, Books (refereed)
[1] Kato, F.: Log
smooth deformation theory. Tohoku Math. Journal, 48 (1996), 317354.
[2] Kato, F.: The
relative log Poincare lemma and relative log de Rham theory. Duke Math. Journal, 93 No.1 (1998), 179206.
[3] Kato, F.: Functors
of log Artin rings. Manuscripta Mathematica, 96 (1998), 97112.
[4] Ishida, MN.; Kato,
F.: The strong rigidity theorem for nonarchimedean uniformization. Tohoku Math. Journal, 50 (1998), 537555.
[6] Kato, F.: Log
smooth deformation and moduli of log smooth curves. International Journal
of Math., 11 (2000), 215232.
[7] Kato, F.: Mumford
curves in a specialized pencil of sextics. Manuscripta
Mathematica, 104 (2001), 451458.
[8] Cornelissen, G.; Kato,
F.; Kontogeorgis, A.: Discontinuous groups in positive characteristic
and automorphisms of Mumford curves. Math. Ann., 320 (2001), 5585.
[9] Cornelissen, G.; Kato,
F.: Equivariant deformation of Mumford curves and of ordinary curves in
positive characteristic. Duke Math. Journal, 116 (2003), 431471.
[10] Cornelissen, G.; Kato,
F.: Mumford curves with maximal automorphism group II: Lame type groups
in genus 58. Geom. Dedicata, 102 (2003), 127142.
[11] Cornelissen, G.; Kato,
F.: Mumford curves with maximal automorphism group. Proc. Amer. Math.
Soc., 132 (2004), 19371941.
[12] Kato, F.: Nonarchimedean
orbifolds covered by Mumford curves. Journal of Algebraic
Geom., 14 (2005), 134.
[13] Cornelissen, G.; Kato,
F.: Zur Entartung schwach verzweigter Gruppenoperationen auf Kurven. J. Reine Angew. Math.,
589 (2005), 201236.
[15] Inoue, N.; Kato,
F.: On the Geometry of Wiman's Sextic. J. Math. Kyoto Univ., 45, No.4, (2005) 743757.
[16] Kato, F.;
Ochiai, H.: Arithmetic structure of CMSZ fake projective plane. Journal of Algebra, 305 (2006), 11661185.
[19] Kato, F.: On
the Shimura variety having Mumford's fake projective plane as a connected
component. Math. Z., 259 (2008), 631641.
[20] Cornelissen, G.; Kato,
F.; Kontogeorgis, A.: The relation between rigidanalytic and algebraic
deformation parameters for ArtinSchreierMumford curves. Israel J. Math., 180 (2010), 345370.
[22] Fujiwara, K.;
Gabber, O.; Kato, F.: On Hausdorff completions of commutative rings
in rigid geometry. Journal of Algebra, 322 (2011), 293321.
[23] Byszewski, J.;
Cornelissen, G.; Kato, F.: Un anneau de déformation universel
en conducteur supérieur. Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 2, 2527.
[24] Allcock, D.; Kato,
F.: The densest lattices in PGL3Q2. Adv. Math. 242 (2013), 102115.
[25] Fujiwara, K.; Kato,
F.: Foundations of Rigid Geometry. A book to appear in Monographs in Mathematics, European Mathematical Society Publishing House.
[26] Cornelissen, G.; Kato,
F.; Kool, J.: A combinatorial LiYau inequality and rational points on
curves. To appear in Math. Ann.
Proceeding reports, surveys (refereed)
[14] Cornelissen, G.; Kato,
F.: The padic icosahedron. Notices Amer. Math. Soc., 52, No.7 (2005), 720727.
[17] Fujiwara, K.; Kato,
F.: Rigid Geometry and Applications. In Moduli Spaces and
Arithmetic Geometry (Kyoto, 2004), Advanced Studies in Pure Mathematics, 45 (2006), 325384.
[18] Kato, F.: Rigid
analytic geometry. Translation of Sugaku, 55 (2003), no.4, 392417. Sugaku
Expositions, 20 (2007), no.1, 6595
[21] Kato, F.: Topological
rings in rigid geometry. Motivic integration and its interactions with model theory
and nonArchimedean geometry. Volume I, 103144, London Math. Soc. Lecture Note
Ser., 383, Cambridge Univ.
Press, Cambridge, 2011.
Proceeding reports, surveys (nonrefereed)
[5] Kato, F.: Introduction
to rigid geometry (Japanese), Surikaisekikenkyujo Koukyuroku, 1073 (1998), 148.
[27] Kato, F.; Kontogeorgis,
A.; Kool, J.: On the Galois module structure of polydifferentials of certain
Mumford curves, modular and integral representation theory. (preprint 2012).