Segre S Reflexivity And An Inductive Characterization Of Hyperquadrics
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Segre's Reflexivity and an Inductive Characterization of Hyperquadrics by Yasuyuki Kachi,Eiichi Sato Pdf
Introduction The universal pseudo-quotient for a family of subvarieties Normal bundles of quadrics in $X$ Morphisms from quadrics to Grassmannians Pointwise uniform vector bundles on non-singular quadrics Theory of extensions of families over Hilbert schemes Existence of algebraic quotient--proof of Theorem 0.3 Appendix. Deformations of vector bundles on infinitesimally rigid projective varieties with null global $i$-forms References
Segre's Reflexivity and an Inductive Characterization of Hyperquadrics by Yasuyuki Kachi,Susumu Ariki Pdf
Introduction The universal pseudo-quotient for a family of subvarieties Normal bundles of quadrics in $X$ Morphisms from quadrics to Grassmannians Pointwise uniform vector bundles on non-singular quadrics Theory of extensions of families over Hilbert schemes Existence of algebraic quotient--proof of Theorem 0.3 Appendix. Deformations of vector bundles on infinitesimally rigid projective varieties with null global $i$-forms References.
Anisotropic Hardy Spaces and Wavelets by Marcin Bownik Pdf
Investigates the anisotropic Hardy spaces associated with very general discrete groups of dilations. This book includes the classical isotropic Hardy space theory of Fefferman and Stein and parabolic Hardy space theory of Calderon and Torchinsky.
$\mathcal {R}$-Boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type by Robert Denk,Matthias Hieber,Jan Prüss Pdf
The property of maximal $L_p$-regularity for parabolic evolution equations is investigated via the concept of $\mathcal R$-sectorial operators and operator-valued Fourier multipliers. As application, we consider the $L_q$-realization of an elliptic boundary value problem of order $2m$ with operator-valued coefficients subject to general boundary conditions. We show that there is maximal $L_p$-$L_q$-regularity for the solution of the associated Cauchy problem provided the top order coefficients are bounded and uniformly continuous.
Radially Symmetric Patterns of Reaction-diffusion Systems by Arnd Scheel Pdf
In this paper, bifurcations of stationary and time-periodic solutions to reaction-diffusion systems are studied. We develop a center-manifold and normal form theory for radial dynamics which allows for a complete description of radially symmetric patterns. In particular, we show the existence of localized pulses near saddle-nodes, critical Gibbs kernels in the cusp, focus patterns in Turing instabilities, and active or passive target patterns in oscillatory instabilities.
Invariants of Boundary Link Cobordism by Desmond Sheiham Pdf
An $n$-dimensional $\mu$-component boundary link is a codimension $2$ embedding of spheres $L=\sqcup_{\mu}S^n \subset S^{n+2}$ such that there exist $\mu$ disjoint oriented embedded $(n+1)$-manifolds which span the components of $L$. An $F_\mu$-link is a boundary link together with a cobordism class of such spanning manifolds. The $F_\mu$-link cobordism group $C_n(F_\mu)$ is known to be trivial when $n$ is even but not finitely generated when $n$ is odd. Our main result is an algorithm to decide whether two odd-dimensional $F_\mu$-links represent the same cobordism class in $C_{2q-1}(F_\mu)$ assuming $q>1$. We proceed to compute the isomorphism class of $C_{2q-1}(F_\mu)$, generalizing Levine's computation of the knot cobordism group $C_{2q-1}(F_1)$.Our starting point is the algebraic formulation of Levine, Ko and Mio who identify $C_{2q-1}(F_\mu)$ with a surgery obstruction group, the Witt group $G^{(-1)^q,\mu}(\Z)$ of $\mu$-component Seifert matrices. We obtain a complete set of torsion-free invariants by passing from integer coefficients to complex coefficients and by applying the algebraic machinery of Quebbemann, Scharlau and Schulte. Signatures correspond to 'algebraically integral' simple self-dual representations of a certain quiver (directed graph with loops). These representations, in turn, correspond to algebraic integers on an infinite disjoint union of real affine varieties. To distinguish torsion classes, we consider rational coefficients in place of complex coefficients, expressing $G^{(-1)^q,\mu}(\mathbb{Q})$ as an infinite direct sum of Witt groups of finite-dimensional division $\mathbb{Q}$-algebras with involution.The Witt group of every such algebra appears as a summand infinitely often. The theory of symmetric and hermitian forms over these division algebras is well-developed. There are five classes of algebras to be considered; complete Witt invariants are available for four classes, those for which the local-global principle applies. An algebra in the fifth class, namely a quaternion algebra with non-standard involution, requires an additional Witt invariant which is defined if all the local invariants vanish.
Yang-Mills Measure on Compact Surfaces by Thierry Lévy Pdf
In this memoir we present a new construction and new properties of the Yang-Mills measure in two dimensions. This measure was first introduced for the needs of quantum field theory and can be described informally as a probability measure on the space of connections modulo gauge transformations on a principal bundle. We consider the case of a bundle over a compact orientable surface. Our construction is based on the discrete Yang-Mills theory of which we give a full acount. We are able to take its continuum limit and to define a pathwise multiplicative process of random holonomy indexed by the class of piecewise embedded loops. We study in detail the links between this process and a white noise and prove a result of asymptotic independence in the case of a semi-simple structure group. We also investigate global Markovian properties of the measure related to the surgery of surfaces.
Shock-Wave Solutions of the Einstein Equations with Perfect Fluid Sources: Existence and Consistency by a Locally Inertial Glimm Scheme by Jeff Groah,Blake Temple Pdf
Demonstrates the consistency of the Einstein equations at the level of shock-waves by proving the existence of shock wave solutions of the spherically symmetric Einstein equations for a perfect fluid, starting from initial density and velocity profiles that are only locally of bounded total variation.
Kahler Spaces, Nilpotent Orbits, and Singular Reduction by Johannes Huebschmann Pdf
For a stratified symplectic space, a suitable concept of stratified Kahler polarization encapsulates Kahler polarizations on the strata and the behaviour of the polarizations across the strata and leads to the notion of stratified Kahler space which establishes an intimate relationship between nilpotent orbits, singular reduction, invariant theory, reductive dual pairs, Jordan triple systems, symmetric domains, and pre-homogeneous spaces: the closure of a holomorphic nilpotent orbit or, equivalently, the closure of the stratum of the associated pre-homogeneous space of parabolic type carries a (positive) normal Kahler structure. In the world of singular Poisson geometry, the closures of principal holomorphic nilpotent orbits, positive definite hermitian JTS', and certain pre-homogeneous spaces appear as different incarnations of the same structure.The closure of the principal holomorphic nilpotent orbit arises from a semisimple holomorphic orbit by contraction. Symplectic reduction carries a positive Kahler manifold to a positive normal Kahler space in such a way that the sheaf of germs of polarized functions coincides with the ordinary sheaf of germs of holomorphic functions. Symplectic reduction establishes a close relationship between singular reduced spaces and nilpotent orbits of the dual groups.Projectivization of holomorphic nilpotent orbits yields exotic (positive) stratified Kahler structures on complex projective spaces and on certain complex projective varieties including complex projective quadrics. The space of (in general twisted) representations of the fundamental group of a closed surface in a compact Lie group or, equivalently, a moduli space of central Yang-Mills connections on a principal bundle over a surface, inherits a (positive) normal (stratified) Kahler structure. Physical examples are provided by certain reduced spaces arising from angular momentum zero.
Exceptional Vector Bundles, Tilting Sheaves and Tilting Complexes for Weighted Projective Lines by Hagen Meltzer,Richard D. Canary,Darryl McCullough Pdf
This work deals with weighted projective lines, a class of non-commutative curves modelled by Geigle and Lenzing on a graded commutative sheaf theory. They play an important role in representation theory of finite-dimensional algebras; the complexity of the classification of coherent sheaves largely depends on the genus of these curves. We study exceptional vector bundles on weighted projective lines and show in particular that the braid group acts transitively on the set of complete exceptional sequences of such bundles. We further investigate tilting sheaves on weighted projective lines and determine the Auslander-Reiten components of modules over their endomorphism rings. Finally we study tilting complexes in the derived category and present detailed classification results in the case of weighted projective lines of hyperelliptic type.
The Role of the Spectrum in the Cyclic Behavior of Composition Operators by Eva A. Gallardo-Gutieŕrez,Alfonso Montes-Rodríguez Pdf
Introduction and preliminaries Linear fractional maps with an interior fixed point Non elliptic automorphisms The parabolic non automorphism Supercyclic linear fractional composition operators Endnotes Bibliography.
Noether-Lefschetz Problems for Degeneracy Loci by Jeroen Spandaw Pdf
In this monograph we study the cohomology of degeneracy loci of the following type. Let $X$ be a complex projective manifold of dimension $n$, let $E$ and $F$ be holomorphic vector bundles on $X$ of rank $e$ and $f$, respectively, and let $\psi\colon F\to E$ be a holomorphic homomorphism of vector bundles. Consider the degeneracy locus $Z:=D_r(\psi):=\{x\in X\colon \mathrm{rk} (\psi(x))\le r\}.$ We assume without loss of generality that $e\ge f >r\ge 0$. We assume furthermore that $E\otimes F^\vee$ is ample and globally generated, and that $\psi$ is a general homomorphism. Then $Z$ has dimension $d:=n-(e-r)(f-r)$. In order to study the cohomology of $Z$, we consider the Grassmannian bundle $\pi\colon Y:=\mathbb{G}(f-r,F)\to X$ of $(f-r)$-dimensional linear subspaces of the fibres of $F$. In $Y$ one has an analogue $W$ of $Z$: $W$ is smooth and of dimension $d$, the projection $\pi$ maps $W$ onto $Z$ and $W\stackrel{\sim}{\to} Z$ if $n(e-r+1)(f-r+1)$. (If $r=0$ then $W=Z\subseteq X=Y$ is the zero-locus of $\psi\in H^0(X,E\otimes F^\vee)$.) Fulton and Lazarsfeld proved that $ H^q(Y;\mathbb{Z}) \to H^q(W;\mathbb{Z}) $ is an isomorphism for $q
On the Geometry of Some Special Projective Varieties by Francesco Russo Pdf
Providing an introduction to both classical and modern techniques in projective algebraic geometry, this monograph treats the geometrical properties of varieties embedded in projective spaces, their secant and tangent lines, the behavior of tangent linear spaces, the algebro-geometric and topological obstructions to their embedding into smaller projective spaces, and the classification of extremal cases. It also provides a solution of Hartshorne’s Conjecture on Complete Intersections for the class of quadratic manifolds and new short proofs of previously known results, using the modern tools of Mori Theory and of rationally connected manifolds. The new approach to some of the problems considered can be resumed in the principle that, instead of studying a special embedded manifold uniruled by lines, one passes to analyze the original geometrical property on the manifold of lines passing through a general point and contained in the manifold. Once this embedded manifold, usually of lower codimension, is classified, one tries to reconstruct the original manifold, following a principle appearing also in other areas of geometry such as projective differential geometry or complex geometry.
Numerical Control over Complex Analytic Singularities by David B. Massey Pdf
The Milnor number is a powerful invariant of an isolated, complex, affine hyper surface singularity. It provides data about the local, ambient, topological-type of the hyper surface, and the constancy of the Milnor number throughout a family implies that Thom's $a_f$ condition holds and that the local, ambient, topological-type is constant in the family. Much of the usefulness of the Milnor number is due to the fact that it can be effectively calculated in an algebraic manner.The Le cycles and numbers are a generalization of the Milnor number to the setting of complex, affine hyper surface singularities, where the singular set is allowed to be of arbitrary dimension. As with the Milnor number, the Le numbers provide data about the local, ambient, topological-type of the hyper surface, and the constancy of the Le numbers throughout a family implies that Thom's $a_f$ condition holds and that the Milnor fibrations are constant throughout the family. Again, much of the usefulness of the Le numbers is due to the fact that they can be effectively calculated in an algebraic manner.In this work, we generalize the Le cycles and numbers to the case of hyper surfaces inside arbitrary analytic spaces. We define the Le-Vogel cycles and numbers, and prove that the Le-Vogel numbers control Thom's $a_f$ condition. We also prove a relationship between the Euler characteristic of the Milnor fibre and the Le-Vogel numbers. Moreover, we give examples which show that the Le-Vogel numbers are effectively calculable. In order to define the Le-Vogel cycles and numbers, we require, and include, a great deal of background material on Vogel cycles, analytic intersection theory, and the derived category. Also, to serve as a model case for the Le-Vogel cycles, we recall our earlier work on the Le cycles of an affine hyper surface singularity.
