Elliptic three-folds II: Multiple fibres
Mark
Gross
3409-3468
Abstract: Let $f:X\rightarrow S$ be an elliptic fibration with a section, where $S$ is a projective surface and $X$ is a projective threefold. We determine when it is possible to perform a logarithmic transformation along a closed subset $Z\subseteq S$ to obtain a new elliptic fibration $f':X'\rightarrow S$ which now has multiple fibres along $Z$. This is done in the setting of Ogg-Shafarevich theory. We find a number of obstructions to performing such a logarithmic transformation, the very last of which takes values in the torsion part of the codimension 2 Chow group of $X$.
On perfect isometries and isotypies in alternating groups
Paul
Fong;
Morton
E.
Harris
3469-3516
Abstract: Perfect isometries and isotypies are constructed for alternating groups between blocks with abelian defect groups and the Brauer correspondents of these blocks. These perfect isometries and isotypies satisfy additional compatibility conditions which imply that an extended Broué conjecture holds for the principal block of an almost simple group with an abelian Sylow $p$-subgroup and a generalized Fitting subgroup isomorphic to an alternating group.
Monoid Hecke algebras
Mohan
S.
Putcha
3517-3534
Abstract: This paper concerns the monoid Hecke algebras $\mathcal {H}$ introduced by Louis Solomon. We determine explicitly the unities of the orbit algebras associated with the two-sided action of the Weyl group $W$. We use this to: find a description of the irreducible representations of $\mathcal {H}$, find an explicit isomorphism between $\mathcal {H}$ and the monoid algebra of the Renner monoid $R$, extend the Kazhdan-Lusztig involution and basis to $\mathcal {H}$, and prove, for a $W\times W$ orbit of $R$, the existence (conjectured by Renner) of generalized Kazhdan-Lusztig polynomials.
Endomorphism algebras of peak $I$-spaces over posets of infinite prinjective type
Rüdiger
Göbel;
Warren
May
3535-3567
Abstract: We will derive a general result for $R$-categories which allows us to derive the existence of large objects with prescribed endomorphism algebras from the existence of small families. This theorem is based on an earlier result of S. Shelah in which he established the existence of indecomposable abelian groups of any cardinality. We will apply this `Shelah-elevator' for abelian groups and - which is our main concern - for prescribing endomorphism algebras of peak $I$-spaces which are classified by a recent result of Simson.
Spherical functions on symmetric cones
P.
Sawyer
3569-3584
Abstract: In this note, we obtain a recursive formula for the spherical functions associated with the symmetric cone of a formally real Jordan algebra. We use this formula as an inspiration for a similar recursive formula involving the Jack polynomials.
Absolute Borel sets and function spaces
Witold
Marciszewski;
Jan
Pelant
3585-3596
Abstract: An internal characterization of metric spaces which are absolute Borel sets of multiplicative classes is given. This characterization uses complete sequences of covers, a notion introduced by Frolík for characterizing Cech-complete spaces. We also show that the absolute Borel class of $X$ is determined by the uniform structure of the space of continuous functions $C_{p}(X)$; however the case of absolute $G_{\delta }$ metric spaces is still open. More precisely, we prove that, for metrizable spaces $X$ and $Y$, if $\Phi : C_{p}(X) \rightarrow C_{p}(Y)$ is a uniformly continuous surjection and $X$ is an absolute Borel set of multiplicative (resp., additive) class $\alpha$, $\alpha >1$, then $Y$ is also an absolute Borel set of the same class. This result is new even if $\Phi$ is a linear homeomorphism, and extends a result of Baars, de Groot, and Pelant which shows that the \v{C}ech-completeness of a metric space $X$ is determined by the linear structure of $C_{p}(X)$.
Polynomial structures on polycyclic groups
Karel
Dekimpe;
Paul
Igodt
3597-3610
Abstract: We know, by recent work of Benoist and of Burde & Grunewald, that there exist polycyclic-by-finite groups $G$, of rank $h$ (the examples given were in fact nilpotent), admitting no properly discontinuous affine action on $\mathbb {R}^h$. On the other hand, for such $G$, it is always possible to construct a properly discontinuous smooth action of $G$ on $\mathbb {R}^h$. Our main result is that any polycyclic-by-finite group $G$ of rank $h$ contains a subgroup of finite index acting properly discontinuously and by polynomial diffeomorphisms of bounded degree on $\mathbb {R}^h$. Moreover, these polynomial representations always appear to contain pure translations and are extendable to a smooth action of the whole group $G$.
Arithmeticity, discreteness and volume
F.
W.
Gehring;
C.
Maclachlan;
G.
J.
Martin;
A.
W.
Reid
3611-3643
Abstract: We give an arithmetic criterion which is sufficient to imply the discreteness of various two-generator subgroups of ${PSL}(2,\mathbf {c})$. We then examine certain two-generator groups which arise as extremals in various geometric problems in the theory of Kleinian groups, in particular those encountered in efforts to determine the smallest co-volume, the Margulis constant and the minimal distance between elliptic axes. We establish the discreteness and arithmeticity of a number of these extremal groups, the associated minimal volume arithmetic group in the commensurability class and we study whether or not the axis of a generator is simple. We then list all ``small'' discrete groups generated by elliptics of order $2$ and $n$, $n=3,4,5,6,7$.
The nonexistence of expansive homeomorphisms of a class of continua which contains all decomposable circle-like continua
Hisao
Kato
3645-3655
Abstract: A homeomorphism $f:X \to X$ of a compactum $X$ with metric $d$ is expansive if there is $c > 0$ such that if $x, y \in X$ and $x \not = y$, then there is an integer $n \in % \mathbf {Z}$ such that $d(f^{n}(x),f^{n}(y)) > c$. It is well-known that $p$-adic solenoids $S_p$ ($p\geq 2$) admit expansive homeomorphisms, each $S_p$ is an indecomposable continuum, and $S_p$ cannot be embedded into the plane. In case of plane continua, the following interesting problem remains open: For each $1 \leq n \leq 3$, does there exist a plane continuum $X$ so that $X$ admits an expansive homeomorphism and $X$ separates the plane into $n$ components? For the case $n=2$, the typical plane continua are circle-like continua, and every decomposable circle-like continuum can be embedded into the plane. Naturally, one may ask the following question: Does there exist a decomposable circle-like continuum admitting expansive homeomorphisms? In this paper, we prove that a class of continua, which contains all chainable continua, some continuous curves of pseudo-arcs constructed by W. Lewis and all decomposable circle-like continua, admits no expansive homeomorphisms. In particular, any decomposable circle-like continuum admits no expansive homeomorphism. Also, we show that if $f:X\to X$ is an expansive homeomorphism of a circle-like continuum $X$, then $f$ is itself weakly chaotic in the sense of Devaney.
The class number one problem for some non-abelian normal CM-fields
Stéphane
Louboutin;
Ryotaro
Okazaki;
Michel
Olivier
3657-3678
Abstract: Let ${\bf N}$ be a non-abelian normal CM-field of degree $4p,$ $p$ any odd prime. Note that the Galois group of ${\bf N}$ is either the dicyclic group of order $4p,$ or the dihedral group of order $4p.$ We prove that the (relative) class number of a dicyclic CM-field of degree $4p$ is always greater then one. Then, we determine all the dihedral CM-fields of degree $12$ with class number one: there are exactly nine such CM-fields.
Quadratic optimal control of stable well-posed linear systems
Olof
J.
Staffans
3679-3715
Abstract: We consider the infinite horizon quadratic cost minimization problem for a stable time-invariant well-posed linear system in the sense of Salamon and Weiss, and show that it can be reduced to a spectral factorization problem in the control space. More precisely, we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution. If both the system and the spectral factor are regular, then the feedback operator can be expressed in terms of the Riccati operator, and the Riccati operator is a positive self-adjoint solution of an algebraic Riccati equation. This Riccati equation is similar to the usual algebraic Riccati equation, but one of its coefficients varies depending on the subspace in which the equation is posed. Similar results are true for unstable systems, as we have proved elsewhere.
Asymptotic behaviour of reproducing kernels of weighted Bergman spaces
Miroslav
Englis
3717-3735
Abstract: Let $\Omega$ be a domain in $\mathbb {C}^{n}$, $F$ a nonnegative and $G$ a positive function on $\Omega$ such that $1/G$ is locally bounded, $A^{2}_{\alpha }$ the space of all holomorphic functions on $\Omega$ square-integrable with respect to the measure $F^{\alpha }G\,d\lambda$, where $d\lambda$ is the $2n$-dimensional Lebesgue measure, and $K_{\alpha }(x,y)$ the reproducing kernel for $A^{2}_{\alpha }$. It has been known for a long time that in some special situations (such as on bounded symmetric domains $\Omega$ with $G=\text {\bf 1}$ and $F=\,$the Bergman kernel function) the formula \begin{equation*}\lim _{\alpha \to +\infty }K_{\alpha }(x,x)^{1/\alpha }=1/F(x) \tag {$*$} \end{equation*} holds true. [This fact even plays a crucial role in Berezin's theory of quantization on curved phase spaces.] In this paper we discuss the validity of this formula in the general case. The answer turns out to depend on, loosely speaking, how well the function $-\log F$ can be approximated by certain pluriharmonic functions lying below it. For instance, ($*$) holds if $-\log F$ is convex (and, hence, can be approximated from below by linear functions), for any function $G$. Counterexamples are also given to show that in general ($*$) may fail drastically, or even be true for some $x$ and fail for the remaining ones. Finally, we also consider the question of convergence of $K_{\alpha }(x,y)^{1/\alpha }$ for $x\neq y$, which leads to an unexpected result showing that the zeroes of the reproducing kernels are affected by the smoothness of $F$: for instance, if $F$ is not real-analytic at some point, then $K_{\alpha }(x,y)$ must have zeroes for all $\alpha$ sufficiently large.
The Brauer group of Yetter-Drinfel'd module algebras
S.
Caenepeel;
F.
Van
Oystaeyen;
Y.
H.
Zhang
3737-3771
Abstract: Let $H$ be a Hopf algebra with bijective antipode. In a previous paper, we introduced $H$-Azumaya Yetter-Drinfel'd module algebras, and the Brauer group ${\mathrm {BQ}}(k,H)$ classifying them. We continue our study of ${\mathrm {BQ}}(k,H)$, and we generalize some properties that were previously known for the Brauer-Long group. We also investigate separability properties for $H$-Azumaya algebras, and this leads to the notion of strongly separable $H$-Azumaya algebra, and to a new subgroup of the Brauer group ${\mathrm {BQ}}(k,H)$.
Boundary limits and non-integrability of $\mathcal M$-subharmonic functions in the unit ball of $\mathbb C^n (n\ge 1)$
Manfred
Stoll
3773-3785
Abstract: In this paper we consider weighted non-tangential and tangential boundary limits of non-negative functions on the unit ball $B$ in ${{\mathbb {C}}^{\vphantom {P}}}^{n}$ that are subharmonic with respect to the Laplace-Beltrami operator $\widetilde {\varDelta }$ on $B$. Since the operator $\widetilde {\varDelta }$ is invariant under the group $\mathcal {M}$ of holomorphic automorphisms of $B$, functions that are subharmonic with respect to $\widetilde {\varDelta }$ are usually referred to as $\mathcal {M}$-subharmonic functions. Our main result is as follows: Let $f$ be a non-negative $\mathcal {M}$-subharmonic function on $B$ satisfying \begin{equation*}\int _{B} (1-|z|^{2})^{\gamma }f^{p}(z)\,d\lambda (z)< \infty \end{equation*} for some $p> 0$ and some $\gamma >\min \{n,pn\}$, where $\lambda$ is the $\mathcal {M}$-invariant measure on $B$. Suppose $\tau \ge 1$. Then for a.e. $\zeta \in S$, \begin{equation*}f^{p}(z)= o\left ((1-|z|^{2})^{n/\tau -\gamma }\right ) \end{equation*} uniformly as $z\to \zeta$ in each $\mathcal {T}_{\tau ,\alpha }(\zeta )$, where for $\alpha >0$ ($\alpha >\frac {1}{2}$ when $\tau =1$) \begin{equation*}\mathcal {T}_{\tau ,\alpha }(\zeta ) = \{z\in B: |1-\langle z,\zeta \rangle |^{\tau } <\alpha (1-|z|^{2}) \}. \end{equation*} We also prove that for $\gamma \le \min \{n,pn\}$ the only non-negative $\mathcal {M}$-subharmonic function satisfying the above integrability criteria is the zero function.
Line bundle Laplacians over isospectral nilmanifolds
Dorothee
Schueth
3787-3802
Abstract: We show that nontrivial isospectral deformations of a big class of compact Riemannian two-step nilmanifolds can be distinguished from trivial deformations by the behaviour of bundle Laplacians on certain non-flat hermitian line bundles over these manifolds.
A construction of codimension three arithmetically Gorenstein subschemes of projective space
Juan
C.
Migliore;
Chris
Peterson
3803-3821
Abstract: This paper presents a construction method for a class of codimension three arithmetically Gorenstein subschemes of projective space. These schemes are obtained from degeneracy loci of sections of certain specially constructed rank three reflexive sheaves. In contrast to the structure theorem of Buchsbaum and Eisenbud, we cannot obtain every arithmetically Gorenstein codimension three subscheme by our method. However, certain geometric applications are facilitated by the geometric aspect of this construction, and we discuss several examples of this in the final section.
On Frobenius algebras and the quantum Yang-Baxter equation
K.
I.
Beidar;
Y.
Fong;
A.
Stolin
3823-3836
Abstract: It is shown that every Frobenius algebra over a commutative ring determines a class of solutions of the quantum Yang-Baxter equation, which forms a subbimodule of its tensor square. Moreover, this subbimodule is free of rank one as a left (right) submodule. An explicit form of a generator is given in terms of the Frobenius homomorphism. It turns out that the generator is invertible in the tensor square if and only if the algebra is Azumaya.