1.
Nonrational weighted hypersurfaces - Okada, Takuzo
The aim of this paper is to construct (i) infinitely many families of nonrational $\mathbb{Q}$-Fano varieties of arbitrary dimension $\ge 4$ with at most quotient singularities, and (ii) twelve families of nonrational $\mathbb{Q}$-Fano threefolds with at most terminal singularities among which two are new and the remaining ten give an alternate proof of nonrationality to known examples. These are constructed as weighted hypersurfaces with the reduction mod $p$ method introduced by Kollár [10].
2.
A packing problem for holomorphic curves - Tsukamoto, Masaki
We propose a new approach to the value distribution theory of entire holomorphic curves. We define \emph{packing density} of Brody curves, and show that it has various non-trivial properties. The packing density of Brody curves can be considered as an infinite dimensional version of characteristic number, and it has an application to Gromov's mean dimension theory.
3.
On canonical modules of toric face rings - Ichim, Bogdan; Römer, Tim
Generalizing the concepts of Stanley-Reisner and affine monoid algebras, one can associate to a rational pointed fan $\Sigma$ in $\mathbb{R}^{d}$ the $\mathbb{Z}^{d}$-graded toric face ring $K[\Sigma]$. Assuming that $K[\Sigma]$ is Cohen-Macaulay, the main result of this paper is to characterize the situation when its canonical module is isomorphic to a $\mathbb{Z}^{d}$-graded ideal of $K[\Sigma]$. From this result several algebraic and combinatorial consequences are deduced. As an application, we give a relation between the cleanness of $K[\Sigma]$ and the shellability of $\Sigma$.
4.
The absolute Galois group of the field of totally $S$-adic numbers - Haran, Dan; Jarden, Moshe; Pop, Florian
For a finite set $S$ of primes of a number field $K$ and for $\sigma_{1}, \dots, \sigma_{e} \in \operatorname{Gal}(K)$ we denote the field of totally $S$-adic numbers by $K_{{\rm tot}, S}$ and the fixed field of $\sigma_{1}, \dots, \sigma_{e}$ in $K_{{\rm tot}, S}$ by $K_{{\rm tot}, S}({\boldsymbol\sigma})$. We prove that for almost all ${\boldsymbol\sigma} \in \operatorname{Gal}(K)^{e}$ the absolute Galois group of $K_{{\rm tot}, S}({\boldsymbol\sigma})$ is the free product of ${\hat F}_{e}$ and a free product of local factors over $S$.
5.
$C^{\infty}$-convergence of circle patterns to minimal surfaces - Lan, Shi-Yi; Dai, Dao-Qing
Given a smooth minimal surface $F : \Omega \rightarrow \mathbb{R}^{3}$ defined on a simply connected region $\Omega$ in the complex plane $\mathbb{C}$, there is a regular SG circle pattern $Q_{\Omega}^{\epsilon}$. By the Weierstrass representation of $F$ and the existence theorem of SG circle patterns, there exists an associated SG circle pattern $P_{\Omega}^{\epsilon}$ in $\mathbb{C}$ with the combinatoric of $Q_{\Omega}^{\epsilon}$. Based on the relationship between the circle pattern $P_{\Omega}^{\epsilon}$ and the corresponding discrete minimal surface $F^{\epsilon} : V_{\Omega}^{\epsilon} \rightarrow \mathbb{R}^{3}$ defined on the vertex set $V_{\Omega}^{\epsilon}$ of the graph of $Q_{\Omega}^{\epsilon}$, we show that there exists a family of discrete minimal...
6.
Canonical bases of Borcherds-Cartan type - Li, Yiqiang; Lin, Zongzhu
We study the canonical basis for the negative part $\mathbf{U}^{-}$ of the quantum generalized Kac-Moody algebra associated to a symmetric Borcherds-Cartan matrix. The algebras $\mathbf{U}^{-}$ associated to two different matrices satisfying certain conditions may coincide (6.3). We show that the canonical bases coincide provided that the algebras $\mathbf{U}^{-}$ coincide (Theorem 6.3.5). We also answer partially a question by Lusztig in [L3] (Theorem 7.1.1).
7.
Very ampleness of adjoint linear systems on smooth surfaces with boundary - Ma?ek, Vladimir
Let $M$ be a $\Q$-divisor on a smooth surface over $\C\,$. In this paper we give criteria for very ampleness of the adjoint of $\rup{M}$, the round-up of $M$. (Similar results for global generation were given by Ein and Lazarsfeld and used in their proof of Fujita's Conjecture in dimension 3.) In \S 4 we discuss an example which suggests that this kind of criteria might also be useful in the study of linear systems on surfaces.
8.
{${\bf Z}$}-graduations de type parabolique et polynômes harmoniques - Mortajine, Abdel Latif
We give pluriharmonic and harmonic representations of prehomogeneous vector spaces of regular classical parabolic type $(L = \Gamma \times M, V)$ througt the commutative quotients $V \mathbin{{/}\!{/}} M = \mathop{\rm spec} (C[V]^{M})$ of them.
9.
Symmetries in the fourth Painlevé equation and Okamoto polynomials - Noumi, Masatoshi; Yamada, Yasuhiko
The fourth Painlev\'e equation $P_{IV}$ is known to have symmetry of the affine Weyl group of type $A^{(1)}_2$ with respect to the Bäcklund transformations. We introduce a new representation of $P_{IV}$, called the
symmetric form
, by taking the three fundamental invariant divisors as the dependent variables. A complete description of the symmetry of $P_{IV}$ is given in terms of this representation. Through the symmetric form, it turns out that $P_{IV}$ is obtained as a similarity reduction of the 3-reduced modified KP hierarchy. It is proved in particular that the special polynomials for rational solutions $P_{IV}$, called
Okamoto polynomials
, are expressible in terms...
10.
A sufficient condition for Nevanlinna parametrization and an extension of Heins theorem - Takahashi, Sechiko
An extended interpolation problem on a Riemann surface is formulated in terms of local rings and ideals. A sufficient condition for Nevanlinna parametrization is obtained. By means of this, Heins theorem on Pick-Nevanlinna interpolation in doubly connected domains is generalized to extended interpolation.
11.
The martingale problem for pseudo-differential operators on infinite-dimensional spaces - Bogachev, V.; Lescot, P.; Röckner, M.
A martingale problem for pseudo-differential operators on infinite dimensional spaces is formulated and the existence of a solution is proved. Applications to infinite dimensional "stable-like" processes are presented.
12.
Distribution of length spectrum of circles on a complex hyperbolic space - Adachi, Toshiaki
It is well-known that all geodesics on a Riemannian symmetric space of rank one are congruent each other under the action of isometry group. Being concerned with circles, we also know that two closed circles in a real space form are congruent if and only if they have the same length. In this paper we study how prime periods of circles on a complex hyperbolic space are distributed on a real line and show that even if two circles have the same length and the same geodesic curvature they are not necessarily congruent each other.
13.
Componentwise linear ideals - Herzog, Jürgen; Hibi, Takayuki
A componentwise linear ideal is a graded ideal $I$ of a polynomial ring such that, for each degree $q$, the ideal generated by all homogeneous polynomials of degree $q$ belonging to $I$ has a linear resolution. Examples of componentwise linear ideals include stable monomial ideals and Gotzmann ideals. The graded Betti numbers of a componentwise linear ideal can be determined by the graded Betti numbers of its components. Combinatorics on squarefree componentwise linear ideals will be especially studied. It turns out that the Stanley-Reisner ideal $I_{\Delta}$ arising from a simplicial complex $\Delta$ is componentwise linear if and only if the...
14.
Hilbert-Asai Eisenstein series, regularized products, and heat kernels - Jorgenson, Jay; Lang, Serge
In a famous paper, Asai indicated how to develop a theory of Eisenstein series for arbitrary number fields, using hyperbolic 3-space to take care of the complex places. Unfortunately he limited himself to class number 1. The present paper gives a detailed exposition of the general case, to be used for many applications. First, it is shown that the Eisenstein series satisfy the authors'
definition of regularized products satisfying the generalized Lerch formula, and the basic axioms which allow the systematic development of the authors' theory, including the Cramér theorem.
It is indicated how previous results of Efrat and Zograf for the...
15.
Multiple zeta values, poly-Bernoulli numbers, and related zeta functions - Arakawa, Tsuneo; Kaneko, Masanobu
We study the function $$\zeta(k_1,\dots,k_{n-1};s)=\sum_{0
16.
On the Borel summability of divergent solutions of the heat equation - Lutz, D. A.; Miyake, M.; Schäfke, R.
In recent years, the theory of Borel summability or multisummability of divergent power series of one variable has been established and it has been proved that every formal solution of an ordinary differential equation with irregular singular point is multisummable. For partial differential equations the summability problem for divergent solutions has not been studied so well, and in this paper we shall try to develop the Borel summability of divergent solutions of the Cauchy problem of the complex heat equation, since the heat equation is a typical and an important equation where we meet diveregent solutions. In conclusion, the Borel...
17.
Landen inequalities for hypergeometric functions - Qiu, S.-L.; Vuorinen, M.
A generalization of the Landen identity, in the form of an inequality, is proved for hypergeometric functions. Some well-known asymptotic formulas are refined.
18.
On proper holomorphic mappings from domains with {$\bf T$}-action - Coupet, Bernard; Pan, Yifei; Sukhov, Alexandre
We describe the branch locus of a proper holomorphic mapping between two smoothly bounded pseudoconvex domains of finite type in $\cc^2$ under the assumption that the first domain admits a transversal holomorphic action of the unit circle. As an application we show that any proper holomorphic self-mapping of a smoothly bounded pseudoconvex complete circular domain of finite type in $\cc^2$ is biholomorphic.
19.
Some notes on the moduli of stable sheaves on elliptic surfaces - Yoshioka, K?ta
In this paper, we shall consider the birational structure of moduli of stable sheaves on elliptic surfaces, which is a generalization of Friedman's results to higher rank cases. As applications, we show that some moduli spaces of stable sheaves on ${\Bbb P}^2$ are rational. We also compute the Picard groups of those on Abelian surfaces.
1.
