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Publications of the Research Institute for Mathematical Sciences
Publications of the Research Institute for Mathematical Sciences
Liu, Kefeng; Sun, Xiaofeng; Yau, Shing-Tung
Lê, D?ng Tráng
Consider an algebraic map $\Phi:=(f,g):{\mathbb C}^2\rightarrow {\mathbb C}^2$ defined
by the polynomial functions $f$ and $g$ of ${\mathbb C}^2$. In complex dimension $2$
the Jacobian conjecture asserts that, if the determinant $J(f,g)$ of the Jacobian matrix of
$\Phi$ is a non-zero constant, then the algebraic map $\Phi$ is an isomorphism.
Bierstone, Edward; Milman, Pierre D.
Algorithms for resolution of singularities in characteristic zero
are based on Hironaka's idea of reducing the problem to a simpler
question of desingularization of an ``idealistic exponent'' (or
``marked ideal''). How can we determine whether two marked ideals
are equisingular in the sense that they can be resolved by the
same blowing-up sequences? We show there is a desingularization
functor defined on the category of equivalence classes of marked
ideals and smooth morphisms, where marked ideals are ``equivalent''
if they have the same sequences of ``test transformations''.
Functoriality in this sense realizes Hironaka's idealistic exponent
philosophy. We use it to show that the recent algorithms for
desingularization of marked ideals of...
Hayasaka, Norihiko; Yukie, Akihiko
Némethi, András
For any negative definite plumbed 3-manifold $M$ we construct
from its plumbed graph a graded $\Z[U]$-module. This, for rational
homology spheres, conjecturally equals the Heegaard-Floer homology of
Ozsváth and Szabó, but it has even more structure. If $M$ is a
complex singularity link then the normalized Euler-characteristic can
be compared with the analytic invariants. The Seiberg-Witten
Invariant Conjecture is discussed in the light of
this new object.
Oguiso, Keiji
Among other things, we show that Mordell-Weil groups of finitely many different abelian fibrations of a hyperkähler manifold have essentially no relation, as its birational transformation. Precise definition of the terms
``essentially no relation'' will be given in Introduction.
Boucksom, Sébastien; Favre, Charles; Jonsson, Mattias
We extend to higher dimensions some of the valuative analysis of
singularities of plurisubharmonic (psh) functions developed by
the first two authors.
Following Kontsevich and Soibelman we
describe the geometry of the space $\cV$ of all
normalized valuations on $\C[x_1,\dots,x_n]$ centered at the
origin. It is a union of
simplices naturally endowed with an affine structure. Using relative
positivity properties of divisors living on modifications of
$\C^n$ above the origin, we define formal psh functions on $\cV$,
designed to be analogues
of
the usual psh functions. For
bounded formal psh functions on $\cV$, we define a mixed
Monge-Ampère
operator which reflects the intersection theory of divisors
above the origin of $\C^n$.
This operator associates to any $(n-1)$-tuple...
de Fernex, Tommaso; Ein, Lawrence; Ishii, Shihoko
This paper shows a finiteness property of
a divisorial valuation in terms of arcs.
First we show that every divisorial valuation
over an algebraic variety corresponds to an
irreducible closed subset of the arc space.
Then we define the codimension for this subset
and give a formula of the codimension in terms
of ``relative Mather canonical class''.
By using this subset, we prove that a divisorial
valuation is determined by assigning the values of
finite functions. We also have a criterion for a divisorial
valuation to be a monomial valuation by assigning
the values of finite functions.
Kawamata, Yujiro
A result by Birkar-Cascini-Hacon-McKernan together with the
boundedness of length of extremal rays implies that different
minimal models can be connected by a sequence of flops.
Miyaoka, Yoichi
Let $X$ be a minimal projective surface of general type defined over the complex numbers and
let $C \subset X$ be an irreducible curve of geometric genus $g$.
Given a rational number $\alpha \in [0,1]$, we construct an orbibundle $\tilde{\mathcal{E}}_{\alpha}$
associated with the pair $(X,C)$ and establish the Miyaoka-Yau-Sakai inequality for
$\tilde{\mathcal{E}}_{\alpha}$.
By varying the parameter $\alpha$ in the inequality,
we derive several geometric consequences involving the ``canonical degree'' $CK_X$ of $C$.
Specifically we prove the following two results.
(1) If $K_X^2$ is greater than the topological Euler number $c_2(X)$,
then $CK_X$ is uniformly bounded from above by a function of the invariants
$g, K_X^2$ and $c_2(X)$ (an effective version...
Moriwaki, Atsushi
In this paper, we give the rigidity theorem for a log morphism as an extension of a fixed scheme morphism.
Mori, Shigefumi; Prokhorov, Yuri
A $\mathbb Q$-conic bundle is a proper morphism from a threefold
with only terminal singularities to a normal surface such that
fibers are connected and the anti-canonical divisor is
relatively ample. We study the structure of $\mathbb Q$-conic
bundles near their singular fibers.
One corollary to our main results is that the base surface of
every $\mathbb Q$-conic bundle has only Du Val singularities of
type A (a positive solution of a conjecture by Iskovskikh).
We obtain the complete classification of $\mathbb Q$-conic
bundles under the additional assumption that the singular fiber
is irreducible and the base surface is singular.
Namikawa, Yoshinori
Feigin, B.; Feigin, E.; Jimbo, M.; Miwa, T.; Takeyama, Y.
The filtration of the Virasoro minimal series representations
$M^{(p,p')}_{r,s}$
induced by the $(1,3)$-primary field $\phi_{1,3}(z)$
is studied. For $1< p'/p< 2$, a conjectural basis of $M^{(p,p')}_{r,s}$
compatible with the filtration is given by using monomial vectors in terms of
the Fourier coefficients of $\phi_{1,3}(z)$. In support of this conjecture,
we give two results. First, we establish the equality of the character
of the conjectural basis vectors with the character of the whole representation
space. Second, for the unitary series ($p'=p+1$), we establish for each $m$
the equality between the character of the degree $m$ monomial basis and the character of the
degree $m$ component in the associated graded module
${\rm gr}(M^{(p,p+1)}_{r,s})$...
Balaji, V.; Kollár, János
We define the notion of holonomy group for a stable vector bundle $F$ on a
variety in terms of the NarasimhanSeshadri unitary representation of its restriction
to curves.
¶ Next we relate the holonomy group to the minimal structure group and to the
decomposition of tensor powers of $F$. Finally we illustrate the principle that either
the holonomy is large or there is a clear geometric reason why it should be small.
Oda, Tadao
In addition to his great accomplishments in mathematical research itself,
Professor Hironaka has been very active since 1980 in trying to promote mathematical
sciences and to bring up young mathematical scientists. His accomplishments
in these respects are considerable as explained below.
Lê, D?ng Tráng; Teissier, Bernard
In this succinct and incomplete presentation of Hironaka's published work up to now, it seems convenient to use a covering according to a few main topics: families of spaces and equisingularity, birational and bimeromorphic geometry, finite determinacy and algebraicity problems, flatness and flattening, real analytic and subanalytic geometry. This order follows roughly the order of publication of the first paper in each topic. One common thread is the frequent use of blowing-ups to simplify the algebraic problems or the geometry. For example, in the theory of subanalytic spaces of $\Rr^n$, Hironaka inaugurated and systematically used this technique, in contrast with...
Mori, Shigefumi; Prokhorov, Yuri
Ihara, Yasutaka
Enomoto, Naoya; Kashiwara, Masaki
In the preceding paper, we formulated a conjecture on the relations between certain classes of irreducible representations of a?ne Hecke algebras of type B and symmetric crystals for $gl?$. In the present paper, we prove the existence of the symmetric crystal and the global basis for $gl?$.