arXiv subjects

Fumiharu Kato

Publications and source records attributed to Fumiharu Kato.

25 records · Page 2Linked to original sources

arXiv2001

Equivariant deformation of Mumford curves and of ordinary curves in positive characteristic

We compute the dimension of the tangent space to, and the Krull dimension of the pro-representable hull of two deformation functors. The first one is the ``algebraic'' deformation functor of an ordinary curve X over a field of positive charateristic with prescribed action of a finite group G, and the data are computed in terms of the ramification behaviour of X -> G\X. The second one is the ``analytic'' deformation functor of a fixed embedding of a finitely generated discrete group N in PGL(2,K) over a non-archimedean valued field K, and the data are computed in terms of the Bass-Serre representation of N via a graph of groups. Finally, if F is a free subgroup of N such that N is contained in the normalizer of F in PGL(2,K), then the Mumford curve associated to F becomes equipped with an action of N/F, and we show that the algebraic functor deforming the latter action coincides with the analytic functor deforming the embedding of N.

math.AG↗

arXiv2000

Arithmetic structure of CMSZ fake projective planes

We prove that the fake projective planes constructed from the diadic discrete group discovered by Cartwright, Mantero, Steger, and Zappa are connected Shimura varieties associated to a certain unitary group. The necessary Shimura data, as well as the unitary group, are explicitly described. We also give a field of definition of these fake projective planes.

math.AG↗

arXiv2000

Graph Theoretic Construction of Discrete Groups over p-adic Fields

This paper is originally designed as a part of revision of the author's preprint math.AG/9908174 "P-adic Schwarzian triangle groups of Mumford type". Recently, Yves Andr'e pointed out a flaw in that preprint; more precisely, Proposition II there was not correct, and consequently, as he pointed out, some triangle groups are missing (but only in p=2,3,5). I apologize to the readers, and will release the other part soon by the original title. In this paper, we concentrate on the constructive part, but more systematically and in more general situation. More precisely, this paper presents a complete criterion for an abstract tree of groups to be realized in the context of p-adic uniformization. This criterion provides a practical way of constructing finitely generated discrete subgroups in PGL(2,K) over a p-adic field K. In the end of this paper, we will exhibit concrete examples of such construction, which in particular shows that there exist infinitely many triangle groups.

math.AG↗

arXiv2000

P-adic Schwarzian triangle groups of Mumford type

In this article we discuss a certain p-adic analogue of classical Schwarzian triangle groups, an analogue which is related to Mumford's uniformization of p-adic analytic curves. p-adic Schwarzian triangle groups are defined to be the Galois groups of analytic coverings over projective line with precisely 3 branch points. We say that a p-adic triangle group is of Mumford type if the corresponding universal covering is given by a certain locally compact analytic subspace in projective line, related to Mumford's uniformization. The main theorem provides a complete classification of p-adic triangle groups of Mumford type; our list of these groups contains some of the arithmetic p-adic triangle groups which are discussed by Yves Andre. Notably, we deduce that p-adic triangle groups of Mumford type exist only if p=2,3,5.

math.AG↗

arXiv1999

Arithmetic structure of Mumford's fake projective plane

This is a revised version of a part of the author's preprint "On p-adic uniformization of fake projective planes" (preprint, Max-Planck-Institut fuer Mathematik, 1998 (121)). In this paper we construct explicitly a Shimura surface of PEL-type, associated to a certain unitary group, whose connected components are isomorphic to Mumford's fake projective plane. We also give the canonical model, Shimura field and number of connected components of this Shimura surface. As a consequence of the construction we prove that the Mumford's fake projective plane is an arithmetic complex unit-ball quotient, and has the 7-th cyclotomic field as a field of definition. In Tohoku Math. J. 50 (1998), 537-555, Masanori Ishida and the author discussed two other possible fake projective planes. The author has also obtained the corresponding results as above for these fake projective planes and has written them in the above preprint, which is available in the web page: http://www.mpim-bonn.mpg.de/html/preprints/preprints.html

math.AG↗

arXiv1995

Logarithmic Embeddings and Logarithmic Semistable Reductions

In this paper, we give a criterion for the existence of logarithmic embeddings -- which was first introduced by Steenbrink -- for general normal crossing varieties. Using this criterion, we also give a new proof of the theorem of Kawamata--Namikawa which states a criterion for the existence of the log structures of semistable type.

alg-geom↗

arXiv1994

Log Smooth Deformation Theory

This paper gives a foundation of log smooth deformation theory. We study the infinitesimal liftings of log smooth morphisms and show that the log smooth deformation functor has a representable hull. This deformation theory gives, for example, the following two types of deformations: (1) relative deformations of a certain kind of a pair of an algebraic variety and a divisor of it, and (2) global smoothings of normal crossing varieties. The former is a generalization of the relative deformation theory introduced by Makio, and the latter coincides with the logarithmic deformation theory introduced by Kawamata and Namikawa.

alg-geom↗