The kinematic formula in complex integral geometry
Theodore
Shifrin
255-293
Abstract: Given two nonsingular projective algebraic varieties $X,Y \subset {{\mathbf{P}}^n}$, $Y \subset {{\mathbf{P}}^n}$ meeting transversely, it is classical that one may express the Chern classes of their intersection $X \cap Y$ in terms of the Chern classes of $X$ and $Y$ and the Kähler form (hyperplane class) of $ {{\mathbf{P}}^n}$. This depends on global considerations. However, by putting a hermitian connection on the tangent bundle of $X$, we may interpret the Chern classes of $ X$ as invariant polynomials in the curvature form of the connection. Armed with this local formulation of Chern classes, we now consider two complex submanifolds (not necessarily compact) $ X$, $Y \subset {{\mathbf{P}}^n}$, and investigate the geometry of their intersection. The pointwise relation between the Chern forms of $ X \cap Y$ and those of $ X$ and $Y$ is rather complicated. However, when we average integrals of Chern forms of $X \cap gY$ over all elements $ g$ of the group of motions of $ {{\mathbf{P}}^n}$, these can be expressed in a universal fashion in terms of integrals of Chern forms of $X$ and $Y$. This is, then, the kinematic formula for the unitary group.
Periods of iterated integrals of holomorphic forms on a compact Riemann surface
Shu Yin
Hwang Ma
295-300
Abstract: Holomorphic forms are integrated iteratedly along paths in a compact Riemann surface $M$ of genus $g$, thus inducing a homomorphism from the fundamental group $\Gamma = {\pi _1}(M,{P_0})$ to a proper multiplicative subgroup $G$ of the group of units in $\widehat{T({\Omega ^{1 \ast }})}$, where ${\Omega ^1}$ denotes the space of holomorphic forms on $T$ is the complex dual of $ {\Omega ^1}$, $ T$ means the associated tensor algebra and 11$\hat{ }$'' means completion with respect to the natural grading. The associated homomorphisms from $ \Gamma /{\Gamma ^{(n + 1)}}$ to $G/{G^{(n + 1)}}$ reduces to the classical case ${H_1}(M) \to {\Omega ^{1 \ast }}$ when $ n = 1$. We show that the images of $ \Gamma /{\Gamma ^{(n + 1)}}$ are always cocompact in $G/{G^{(n + 1)}}$ and are discrete for all $n \geqslant 2$ if and only if the Jacobian variety $ J(M)$ of $M$ is isogenous to ${E^g}$ for some elliptic curve $ E$ with complex multiplication.
On the structure of equationally complete varieties. II
Don
Pigozzi
301-319
Abstract: Each member $\mathcal{V}$ of a large family of nonassociative or, when applicable, nondistributive varieties has the following universal property: Every variety $\mathcal{K}$ that satisfies certain very weak versions of the amalgamation and joint embedding properties is isomorphic, as a category, to a coreflective subcategory of some equationally complete subvariety of $ \mathcal{V}$. Moreover, the functor which serves to establish the isomorphism preserves injections. As a corollary one obtains the existence of equationally complete subvarieties of $\mathcal{V}$ that fail to have the amalgamation property and fail to be residually small. The family of varieties universal in the above sense includes commutative groupoids, bisemigroups (i.e., algebras with two independent associative operations), and quasi-groups.
The C. Neumann problem as a completely integrable system on an adjoint orbit
Tudor
Raţiu
321-329
Abstract: It is shown by purely Lie algebraic methods that the ${\text{C}}$. Neumann problem--the motion of a material point on a sphere under the influence of a quadratic potential--is a completely integrable system of Euler-Poisson equations on a minimal-dimensional orbit of a semidirect product of Lie algebras.
Real submanifolds of codimension two in complex manifolds
Hon Fei
Lai
331-352
Abstract: The equivalence problem for a real submanifold $M$ of dimension at least eight and codimension two in a complex manifold is solved under a certain nondegeneracy condition on the Levi form. If the Levi forms at all points of $M$ are equivalent, a normalized Cartan connection can be defined on a certain principal bundle over $ M$. The group of this bundle can be described by means of the osculating quartic of $ M$ or the prolongation of the graded Lie algebra of type $ {\mathfrak{g}_2} \oplus {\mathfrak{g}_1}$ defined by the Levi form.
Configurations of surfaces in $4$-manifolds
Patrick M.
Gilmer
353-380
Abstract: We consider collections of surfaces $ \{ {F_i}\}$ smoothly embedded, except for a finite number of isolated singularities, self-intersections, and mutual intersections, in a $4$-manifold $M$. A small $3$-sphere about each exceptional point will intersect these surfaces in a link. If $[{F_i}] \in {H_2}(M)$ are linearly dependent modulo a prime power, we find lower bounds for $ \Sigma$ genus $ ({F_i})$ in terms of the $ [{F_i}]$, and invariants of the links that describe the exceptional points.
Division by holomorphic functions and convolution equations in infinite dimension
J.-F.
Colombeau;
R.
Gay;
B.
Perrot
381-391
Abstract: Let $E$ be a complex complete dual nuclear locally convex space (i.e. its strong dual is nuclear), $ \Omega$ a connected open set in $E$ and $ \mathcal{E}(\Omega )$ the space of the $ {C^\infty }$ functions on $ \Omega$ (in the real sense). Then we show that any element of $\operatorname{Exp} (E')$ in terms of the zero set of their characteristic functions.
Boundary interpolation sets for holomorphic functions smooth to the boundary and BMO
Joaquim
Bruna
393-409
Abstract: Let ${A^p}$ denote the class of holomorphic functions on the unit disc whose first $p$-derivatives belong to the disc algebra. We characterize the boundary interpolation sets for ${A^p}$, that is, those closed sets $E \subset T$ such that every function in $ {C^p}(E)$ extends to a function in ${A^p}$. We also give a constructive proof of the corresponding result for $ {A^\infty }$ (see [1]). We show that the structure of these sets is in some sense related to BMO and that this fact can be used to obtain precise estimates of outer functions vanishing on $E$.
The $\aleph \sb{2}$-Souslin hypothesis
Richard
Laver;
Saharon
Shelah
411-417
Abstract: We prove the consistency with $CH$ that there are no $ {\aleph _2}$-Souslin trees.
Residually small varieties with modular congruence lattices
Ralph
Freese;
Ralph
McKenzie
419-430
Abstract: We focus on varieties $\mathcal{V}$ of universal algebras whose congruence lattices are all modular. No further conditions are assumed. We prove that if the variety $\mathcal{V}$ is residually small, then the following law holds identically for congruences over algebras in $ \mathcal{V}:\beta \cdot [\delta ,\delta ] \leqslant [\beta ,\delta ]$. (The symbols in this formula refer to lattice operations and the commutator operation defined over any modular variety, by Hagemann and Herrmann.) We prove that a finitely generated modular variety $\mathcal{V}$ is residually small if and only if it satisfies this commutator identity, and in that case $ \mathcal{V}$ is actually residually $< n$ for some finite integer $n$. It is further proved that in a modular variety generated by a finite algebra $ A$ the chief factors of any finite algebra are bounded in cardinality by the size of $A$, and every simple algebra in the variety has a cardinality at most that of $A$.
Stability theorems for the continuous spectrum of a negatively curved manifold
Harold
Donnelly
431-448
Abstract: The spectrum of the Laplacian $\Delta$ for a simply connected complete negatively curved Riemannian manifold is studied. The Laplacian ${\Delta _0}$ of a simply connected constant curvature space ${M_0}$ is known up to unitary equivalence. Decay conditions are given, on the metric $g$ and curvature $K$ of $M$, which imply that the continuous part of ${\Delta _0}$ is unitarily equivalent to ${\Delta _0}$.
Some restrictions on finite groups acting freely on $(S\sp{n})\sp{k}$
Gunnar
Carlsson
449-457
Abstract: Restrictions other than rank conditions on elementary abelian subgroups are found for finite groups acting freely on ${({S^n})^k}$, with trivial action on homology. It is shown that elements $x$ of order $p$, $p$ an odd prime, with $x$ in the normalizer of an elementary abelian $ 2$-subgroup $E$ of $G$, must act trivially on $E$ unless $p\vert(n + 1)$. It is also shown that if $ p = 3$ or $7$, $x$ must act trivially, independent of $ n$. This produces a large family of groups which do not act freely on ${({S^n})^k}$ for any values of $n$ and $k$. For certain primes $p$, using the mod two Steenrod algebra, one may show that $x$ acts trivially unless ${2^{\mu (p)}}\vert(n + 1)$, where $\mu (p)$ is an integer depending on $ p$.
Adjoint operators in Lie algebras and the classification of simple flexible Lie-admissible algebras
Susumu
Okubo;
Hyo Chul
Myung
459-472
Abstract: Let $\mathfrak{A}$ be a finite-dimensional flexible Lie-admissible algebra over an algebraically closed field $F$ of characteristic 0. It is shown that if ${\mathfrak{A}^ - }$ is a simple Lie algebra which is not of type $ {A_n}(n \geqslant 2)$ then $\mathfrak{A}$ is a Lie algebra isomorphic to ${\mathfrak{A}^ - }$, and if ${\mathfrak{A}^ - }$ is a simple Lie algebra of type ${A_n}(n \geqslant 2)$ then $\mathfrak{A}$ is either a Lie algebra or isomorphic to an algebra with multiplication $x \ast y = \mu xy + (1 - \mu )yx - (1/(n + 1))\operatorname{Tr} (xy)I$ which is defined on the space of $(n + 1) \times (n + 1)$ traceless matrices over $F$, where $xy$ is the matrix product and $\mu \ne \frac{1} {2}$ is a fixed scalar in $ F$. This result for the complex field has been previously obtained by employing an analytic method. The present classification is applied to determine all flexible Lie-admissible algebras $ \mathfrak{A}$ such that ${\mathfrak{A}^ - }$ is reductive and the Levi-factor of $ {\mathfrak{A}^ - }$ is simple. The central idea is the notion of adjoint operators in Lie algebras which has been studied in physical literature in conjunction with representation theory.
BP torsion in finite $H$-spaces
Richard
Kane
473-497
Abstract: Let $p$ be odd and $(X,\mu )$ a $1$-connected $ \operatorname{mod} p$ finite $H$-space. It is shown that for $n \geqslant 1$ the Morava $K$-theories, $k{(n)_ \ast }(X)$ and $k{(n)^ \ast }(X)$, have no higher ${\upsilon _n}$ torsion. Also examples are constructed to show that $ {\upsilon _1}$ torsion in $ BP{\langle 1\rangle ^ \ast }(X)$ can be of arbitrarily high order.
Free coverings and modules of boundary links
Nobuyuki
Sato
499-505
Abstract: Let $L = \{ {K_1}, \ldots ,{K_m}\}$ be a boundary link of $n$-spheres in $ {S^{n + 2}}$, where $n \geqslant 3$, and let $X$ be the complement of $L$. Although most of the classical link invariants come from the homology of the universal abelian cover $ \tilde X$ of $ X$, with increasing $ m$ these groups become difficult to manage. For boundary links, there is a canonical free covering $ {X_\omega }$, which is simultaneously a cover of $\tilde X$. Thus, knowledge of ${H_ \ast }{X_\omega }$ yields knowledge of ${H_ \ast }\tilde X$. We study general properties of such covers and obtain, for $ 1 < q < n/2$, a characterization of the groups ${H_q}{X_\omega }$ as modules over the group of covering transformations. Some applications follow.
The Radon-Nikod\'ym property in conjugate Banach spaces. II
Charles
Stegall
507-519
Abstract: In the first part of this article the following result was proved. Theorem. The dual of a Banach space $X$ has the Radon-Nikodym property if and only if for every closed, linear separable subspace $Y$ of $X$, ${Y^ \ast }$ is separable. We find other, more detailed descriptions of Banach spaces whose duals have the Radon-Nikodym property.
On perfect measures
G.
Koumoullis
521-537
Abstract: Let $\mu$ be a nonzero positive perfect measure on a $\sigma$-algebra of subsets of a set $ X$. It is proved that if $\{ {A_i}:i \in I\}$ is a partition of $ X$ with ${\mu ^ \ast }({A_i}) = 0$ for all $i \in I$ and the cardinal of $ I$ non-(Ulam-) measurable, then there is $ J \subset I$ such that $ { \cup _{_{i \in J}}}{A_i}$ is not $\mu$-measurable, generalizing a theorem of Solovay about the Lebesgue measure. This result is used for the study of perfect measures on topological spaces. It is proved that every perfect Borel measure on a metric space is tight if and only if the cardinal of the space is nonmeasurable. The same result is extended to some nonmetric spaces and the relation between perfectness and other smoothness properties of measures on topological spaces is investigated.
Generalized $3$-manifolds whose nonmanifold set has neighborhoods bounded by tori
Matthew G.
Brin
539-555
Abstract: We show that all compact, ANR, generalized $3$-manifolds whose nonmanifold set is 0-dimensional and has a neighborhood system bounded by tori are cell-like images of compact $3$-manifolds if and only if the Poincaré conjecture is true. We also discuss to what extent the assumption of the Poincaré conjecture can be replaced by other hypotheses.
The strong convergence of Schr\"odinger propagators
Alan D.
Sloan
557-570
Abstract: Time dependent versions of the Trotter-Kato theorem are discussed using nonstandard analysis. Both standard and nonstandard results are obtained. In particular, it is shown that if a sequence of generators converges in the strong resolvent topology at each time to a limiting generator and if the sequence of generators and limiting generator uniformly satisfy Kisynski type hypotheses then the corresponding Schrodinger propagators converge strongly. The results are used to analyze time dependent, form bounded perturbations of the Laplacian.
Non-quasi-well behaved closed $\ast $-derivations
Frederick M.
Goodman
571-578
Abstract: Examples are given of a non-quasi-well behaved closed * derivation in $C([0,1] \times [0,1])$ extending the partial derivative, and of a compact subset $ \Omega$ of the plane such that $C(\Omega )$ has no nonzero quasi-well behaved * derivations but $ C(\Omega )$ does admit nonzero closed * derivations.
A short proof of Castelnuovo's criterion of rationality
William E.
Lang
579-582
Abstract: We give a new proof in positive characteristic of Castelnuovo's criterion of rationality of algebraic surfaces. We use crystalline cohomology and the de Rham-Witt complex as a substitute for the transcendeal methods of Kodaira.
A correction and some additions to: ``Reparametrization of $n$-flows of zero entropy'' [Trans. Amer. Math. Soc. {\bf 256} (1979), 289--304; MR 81h:28012]
J.
Feldman;
D.
Nadler
583-585
Abstract: In addition to correcting an error in the previously mentioned paper, we show that if $\upsilon \mapsto {\varphi _w}$ and $w \mapsto {\Psi _\sigma }$ on $X$ and $Y$ are $n$- and $m$-flows, respectively, then the $(n + m)$-flow $ (\upsilon ,w) \mapsto {\varphi _\upsilon } \times {\Psi _w}$ on $X \times Y$ is "loosely Kronecker" if and only if $ \varphi$ and $ \Psi$ are.