Central Kähler metrics
Gideon
Maschler
2161-2182
Abstract: The determinant of the Ricci endomorphism of a Kähler metric is called its central curvature, a notion well-defined even in the Riemannian context. This work investigates two types of Kähler metrics in which this curvature potential gives rise to a potential for a gradient holomorphic vector field. These metric types generalize the Kähler-Einstein notion as well as that of Bando and Mabuchi (1986). Whenever possible the central curvature is treated in analogy with the scalar curvature, and the metrics are compared with the extremal Kähler metrics of Calabi. An analog of the Futaki invariant is employed, both invariants belonging to a family described in the language of holomorphic equivariant cohomology. It is shown that one of the metric types realizes the minimum of an $L^2$ functional defined on the space of Kähler metrics in a given Kähler class. For metrics of constant central curvature, results are obtained regarding existence, uniqueness and a partial classification in complex dimension two. Consequently, on a manifold of Fano type, such metrics and Kähler-Einstein metrics can only exist concurrently. An existence result for the case of non-constant central curvature is stated, and proved in a sequel to this work.
Central Kähler metrics with non-constant central curvature
Andrew
D.
Hwang;
Gideon
Maschler
2183-2203
Abstract: The central curvature of a Riemannian metric is the determinant of its Ricci endomorphism, while the scalar curvature is its trace. A Kähler metric is called central if the gradient of its central curvature is a holomorphic vector field. Such metrics may be viewed as analogs of the extremal Kähler metrics defined by Calabi. In this work, central metrics of non-constant central curvature are constructed on various ruled surfaces, most notably the first Hirzebruch surface. This is achieved via the momentum construction of Hwang and Singer, a variant of an ansatz employed by Calabi (1979) and by Koiso and Sakane (1986). Non-existence, real-analyticity and positivity properties of central metrics arising in this ansatz are also established.
On a measure in Wiener space and applications
K.
S.
Ryu;
M.
K.
Im
2205-2222
Abstract: In this article, we consider a measure in Wiener space, induced by the sum of measures associated with an uncountable set of positive real numbers, and investigate the basic properties of this measure. We apply this measure to the various theories related to Wiener space. In particular, we can obtain a partial answer to Johnson and Skoug's open problems, raised in their 1979 paper. Moreover, we can improve and clarify some theories related to Wiener space.
Contractive projections and operator spaces
Matthew
Neal;
Bernard
Russo
2223-2262
Abstract: Parallel to the study of finite-dimensional Banach spaces, there is a growing interest in the corresponding local theory of operator spaces. We define a family of Hilbertian operator spaces $H_n^k$, $1\le k\le n$, generalizing the row and column Hilbert spaces $R_n$ and $C_n$, and we show that an atomic subspace $X\subset B(H)$ that is the range of a contractive projection on $B(H)$is isometrically completely contractive to an $\ell^\infty$-sum of the $H_n^k$ and Cartan factors of types 1 to 4. In particular, for finite-dimensional $X$, this answers a question posed by Oikhberg and Rosenthal. Explicit in the proof is a classification up to complete isometry of atomic w$^*$-closed $JW^*$-triples without an infinite-dimensional rank 1 w$^*$-closed ideal.
Non-crossing cumulants of type B
Philippe
Biane;
Frederick
Goodman;
Alexandru
Nica
2263-2303
Abstract: We establish connections between the lattices of non-crossing partitions of type B introduced by V. Reiner, and the framework of the free probability theory of D. Voiculescu. Lattices of non-crossing partitions (of type A, up to now) have played an important role in the combinatorics of free probability, primarily via the non-crossing cumulants of R. Speicher. Here we introduce the concept of non-crossing cumulant of type B; the inspiration for its definition is found by looking at an operation of ``restricted convolution of multiplicative functions'', studied in parallel for functions on symmetric groups (in type A) and on hyperoctahedral groups (in type B). The non-crossing cumulants of type B live in an appropriate framework of ``non-commutative probability space of type B'', and are closely related to a type B analogue for the R-transform of Voiculescu (which is the free probabilistic counterpart of the Fourier transform). By starting from a condition of ``vanishing of mixed cumulants of type B'', we obtain an analogue of type B for the concept of free independence for random variables in a non-commutative probability space.
Four-weight spin models and Jones pairs
Ada
Chan;
Chris
Godsil;
Akihiro
Munemasa
2305-2325
Abstract: We introduce and discuss Jones pairs. These provide a generalization and a new approach to the four-weight spin models of Bannai and Bannai. We show that each four-weight spin model determines a ``dual'' pair of association schemes.
An elementary invariant problem and general linear group cohomology restricted to the diagonal subgroup
Marian
F.
Anton
2327-2340
Abstract: Conjecturally, for $p$ an odd prime and $R$ a certain ring of $p$-integers, the stable general linear group $GL(R)$ and the étale model for its classifying space have isomorphic mod $p$ cohomology rings. In particular, these two cohomology rings should have the same image with respect to the restriction map to the diagonal subgroup. We show that a strong unstable version of this last property holds for any rank if $p$ is regular and certain homology classes for $SL_2(R)$ vanish. We check that this criterion is satisfied for $p=3$ as evidence for the conjecture.
Induction theorems of surgery obstruction groups
Masaharu
Morimoto
2341-2384
Abstract: Let $G$ be a finite group. It is well known that a Mackey functor $\{ H \mapsto M(H) \}$ is a module over the Burnside ring functor $\{ H \mapsto \Omega(H) \}$, where $H$ ranges over the set of all subgroups of $G$. For a fixed homomorphism $w : G \to \{ -1, 1 \}$, the Wall group functor $\{ H \mapsto L_n^h ({\mathbb Z}[H], w\vert _H) \}$ is not a Mackey functor if $w$ is nontrivial. In this paper, we show that the Wall group functor is a module over the Burnside ring functor as well as over the Grothendieck-Witt ring functor $\{ H \mapsto {\mathrm{GW}}_0 ({\mathbb Z}, H) \}$. In fact, we prove a more general result, that the functor assigning the equivariant surgery obstruction group on manifolds with middle-dimensional singular sets to each subgroup of $G$ is a module over the Burnside ring functor as well as over the special Grothendieck-Witt ring functor. As an application, we obtain a computable property of the functor described with an element in the Burnside ring.
Finiteness theorems for positive definite $n$-regular quadratic forms
Wai Kiu
Chan;
Byeong-Kweon
Oh
2385-2396
Abstract: An integral quadratic form $f$ of $m$ variables is said to be $n$-regular if $f$ globally represents all quadratic forms of $n$ variables that are represented by the genus of $f$. For any $n \geq 2$, it is shown that up to equivalence, there are only finitely many primitive positive definite integral quadratic forms of $n + 3$variables that are $n$-regular. We also investigate similar finiteness results for almost $n$-regular and spinor $n$-regular quadratic forms. It is shown that for any $n \geq 2$, there are only finitely many equivalence classes of primitive positive definite spinor or almost $n$-regular quadratic forms of $n + 2$ variables. These generalize the finiteness result for 2-regular quaternary quadratic forms proved by Earnest (1994).
On Ramanujan's continued fraction for $(q^2;q^3)_{\infty}/(q;q^3)_{\infty}$
George
E.
Andrews;
Bruce
C.
Berndt;
Jaebum
Sohn;
Ae Ja
Yee;
Alexandru
Zaharescu
2397-2411
Abstract: The continued fraction in the title is perhaps the deepest of Ramanujan's $q$-continued fractions. We give a new proof of this continued fraction, more elementary and shorter than the only known proof by Andrews, Berndt, Jacobsen, and Lamphere. On page 45 in his lost notebook, Ramanujan states an asymptotic formula for a continued fraction generalizing that in the title. The second main goal of this paper is to prove this asymptotic formula.
A positive radial product formula for the Dunkl kernel
Margit
Rösler
2413-2438
Abstract: It is an open conjecture that generalized Bessel functions associated with root systems have a positive product formula for nonnegative multiplicity parameters of the associated Dunkl operators. In this paper, a partial result towards this conjecture is proven, namely a positive radial product formula for the non-symmetric counterpart of the generalized Bessel function, the Dunkl kernel. Radial here means that one of the factors in the product formula is replaced by its mean over a sphere. The key to this product formula is a positivity result for the Dunkl-type spherical mean operator. It can also be interpreted in the sense that the Dunkl-type generalized translation of radial functions is positivity-preserving. As an application, we construct Dunkl-type homogeneous Markov processes associated with radial probability distributions.
Stationary sets for the wave equation in crystallographic domains
Mark
L.
Agranovsky;
Eric
Todd
Quinto
2439-2451
Abstract: Let $W$ be a crystallographic group in $\mathbb R^n$ generated by reflections and let $\Omega$ be the fundamental domain of $W.$ We characterize stationary sets for the wave equation in $\Omega$ when the initial data is supported in the interior of $\Omega.$ The stationary sets are the sets of time-invariant zeros of nontrivial solutions that are identically zero at $t=0$. We show that, for these initial data, the $(n-1)$-dimensional part of the stationary sets consists of hyperplanes that are mirrors of a crystallographic group $\tilde W$, $W<\tilde W.$ This part comes from a corresponding odd symmetry of the initial data. In physical language, the result is that if the initial source is localized strictly inside of the crystalline $\Omega$, then unmovable interference hypersurfaces can only be faces of a crystalline substructure of the original one.
On $L^{p}$ continuity of singular Fourier integral operators
Andrew
Comech;
Scipio
Cuccagna
2453-2476
Abstract: We derive $L^{p}$ continuity of Fourier integral operators with one-sided fold singularities. The argument is based on interpolation of (asymptotics of) $L^{2}$ estimates and $\matheurm{H}^1\to L^1$ estimates. We derive the latter estimates elaborating arguments of Seeger, Sogge, and Stein's 1991 paper. We apply our results to the study of the $L^{p}$ regularity properties of the restrictions of solutions to hyperbolic equations onto timelike hypersurfaces and onto hypersurfaces with characteristic points.
Regularity of weak solutions to the Monge--Ampère equation
Cristian
E.
Gutiérrez;
David
Hartenstine
2477-2500
Abstract: We study the properties of generalized solutions to the Monge-Ampère equation $\det D^2 u = \nu$, where the Borel measure $\nu$ satisfies a condition, introduced by Jerison, that is weaker than the doubling property. When $\nu = f \, dx$, this condition, which we call $D_{\epsilon}$, admits the possibility of $f$ vanishing or becoming infinite. Our analysis extends the regularity theory (due to Caffarelli) available when $0 < \lambda \leq f \leq \Lambda < \infty$, which implies that $\nu = f \, dx$is doubling. The main difference between the $D_{\epsilon}$ case and the case when $f$ is bounded between two positive constants is the need to use a variant of the Aleksandrov maximum principle (due to Jerison) and some tools from convex geometry, in particular the Hausdorff metric.
On Ginzburg's bivariant Chern classes
Shoji
Yokura
2501-2521
Abstract: The convolution product is an important tool in geometric representation theory. Ginzburg constructed the ``bivariant" Chern class operation from a certain convolution algebra of Lagrangian cycles to the convolution algebra of Borel-Moore homology. In this paper we prove a ``constructible function version" of one of Ginzburg's results; motivated by its proof, we introduce another bivariant algebraic homology theory $s\mathbb{AH}$ on smooth morphisms of nonsingular varieties and show that the Ginzburg bivariant Chern class is the unique Grothendieck transformation from the Fulton-MacPherson bivariant theory of constructible functions to this new bivariant algebraic homology theory, modulo a reasonable conjecture. Furthermore, taking a hint from this conjecture, we introduce another bivariant theory $\mathbb{GF}$ of constructible functions, and we show that the Ginzburg bivariant Chern class is the unique Grothendieck transformation from $\mathbb{GF}$ to $s\mathbb{AH}$satisfying the ``normalization condition" and that it becomes the Chern-Schwartz-MacPherson class when restricted to the morphisms to a point.
Construction of $t$-structures and equivalences of derived categories
Leovigildo Alonso
Tarrío;
Ana Jeremías
López;
María José
Souto
Salorio
2523-2543
Abstract: We associate a $t$-structure to a family of objects in $\boldsymbol{\mathsf{D}}(\mathcal{A})$, the derived category of a Grothendieck category $\mathcal{A}$. Using general results on $t$-structures, we give a new proof of Rickard's theorem on equivalence of bounded derived categories of modules. Also, we extend this result to bounded derived categories of quasi-coherent sheaves on separated divisorial schemes obtaining, in particular, Be{\u{\i}}\kern.15emlinson's equivalences.
A $C^1$ function for which the $\omega$-limit points are not contained in the closure of the periodic points
Emma
D'Aniello;
T.
H.
Steele
2545-2556
Abstract: We develop a $C^1$ function $f: [- \frac{1}{6}, 1] \rightarrow [- \frac{1}{6}, 1]$ for which $\Lambda(f) \not= \overline{P(f)}$. This answers a query from Block and Coppel (1992).
Tame sets, dominating maps, and complex tori
Gregery
T.
Buzzard
2557-2568
Abstract: A discrete subset of $\mathbb C^n$ is said to be tame if there is an automorphism of $\mathbb C^n$ taking the given discrete subset to a subset of a complex line; such tame sets are known to allow interpolation by automorphisms. We give here a fairly general sufficient condition for a discrete set to be tame. In a related direction, we show that for certain discrete sets in $\mathbb C^n$ there is an injective holomorphic map from $\mathbb C^n$ into itself whose image avoids an $\epsilon$-neighborhood of the discrete set. Among other things, this is used to show that, given any complex $n$-torus and any finite set in this torus, there exist an open set containing the finite set and a locally biholomorphic map from $\mathbb C^n$ into the complement of this open set.
Fixed points of commuting holomorphic mappings other than the Wolff point
Filippo
Bracci
2569-2584
Abstract: Let $\Delta$ be the unit disc of $\mathbb C$ and let $f,g \in \mathrm{Hol}(\Delta,\Delta)$ be such that $f \circ g = g \circ f$. For $A>1$, let $\mathrm{Fix}_A (f)$. In particular, we prove that $g(\mathrm{Fix}_A (f))\subseteq \mathrm{Fix}_A (f)$. As a consequence, besides conditions for $\mathrm{Fix}_A(f) \cap \mathrm{Fix}_A(g) \neq \emptyset$, we prove a conjecture of C. Cowen in case $f$ and $g$ are univalent mappings.