Narrow Results

Unifying the massive spin-1 field with gravity requires the implementation of a regular vector field that satisfies the spin-1 Proca equation and is a fundamental part of the space–time metric. That vector field is one of the pair of vectors in the line element field $(X,-X)$, which is paramount to the existence of all Lorentzian metrics and Modified General Relativity (MGR). Symmetrization of the spin-1 Klein–Gordon equation in a curved Lorentzian space–time introduces the Lie derivative of the metric along the flow of one of the regular vectors in the line element field. The Proca equation in curved space–time can then be described geometrically in terms of the line element vector, the Lie derivative of the Lorentzian metric and the Ricci tensor, which unifies gravity and the spin-1 field. Related issues concerning charge conservation and the Lorenz constraint, singularities in a spherically symmetric curved space–time and geometrical implications of MGR to quantum theory are discussed. A geometrical unification of gravity with quantum field theory is presented.

Consider a Kleinian singularity ${ℂ}^2/\mathrm{\Gamma }$, where $\mathrm{\Gamma }$ is a finite subgroup of SL₂($ℂ$). In this paper, we introduce a natural stack compactifying the singularity by adding a smooth stacky divisor, and we show that sets of framed sheaves on this stack satisfying certain additional criteria are closely related to a class of Nakajima quiver varieties. This partially extends our previous work on punctual Hilbert schemes of Kleinian singularities.

In this paper, we present a resolution of discrete singular fibers of a closed 5-manifold equipped with a locally free S¹-action, and prove its compatibility with the resolution of cyclic surface singularities in the quotient orbifold by the S¹-action.

The Centre Symmetry Set of a planar curve $M$ is the envelope of affine chords of $M$, i.e. the lines joining points on $M$ with parallel tangent lines. In this paper, we study global geometrical properties of this set including the number of singularities and the number of asymptotes.

Singularities of plane into plane mappings described by parabolic two-component systems of quasi-linear partial differential equations of the first order are studied. Impediments arising in the application of the original Whitney’s approach to such a case are discussed. Hierarchy of singularities is analyzed by the double-scaling expansion method for the simplest $2$-component Jordan system. It is shown that flex is the lowest singularity while higher singularities are given by $(k+1,k+2)$ curves which are of cusp type for $k=2n+1$, $n=1,2,3,\dots .$ Regularization of these singularities by deformation of plane into plane mappings into surface $S^{2+k}(\subset {ℝ}^{2+k})$ to plane is discussed. Applicability of the proposed approach to other parabolic type mappings is noted. We finally compare the results obtained for the parabolic case with non-generic gradient catastrophes for hyperbolic systems.

Let (X, 0) be the germ of a normal space of dimension n+1 with an isolated singularity at 0 and let f be a germ of holomorphic function with an isolated regularity at 0. We prove that the meromorphic extension of the current

We consider the singuralities of 2-dimensional moduli spaces of semi-stable sheaves on k3 surfaces. We show that the moduli space is normal, in particular the siguralities are rational double points. We also describe the exceptional locus on the resolution in terms of exceptional sheaves.

We provide a sketch of the GIT construction of the moduli spaces for the three classes of connections: the class of meromorphic connections with fixed divisor of poles D and its subclasses of integrable and integrable logarithmic connections. We use the Luna Slice Theorem to represent the germ of the moduli space as the quotient of the Kuranishi space by the automorphism group of the central fiber. This method is used to determine the singularities of the moduli space of connections in some examples.

The stable singularities of differential map germs constitute the main source of studying the geometric and topological behavior of these maps. In particular, one interesting problem is to find formulae which allow us to count the isolated stable singularities which appear in the discriminant of a stable deformation of a finitely determined map germ. Mond and Pellikaan showed how the Fitting ideals are related to such singularities and obtain a formula to count the number of ordinary triple points in map germs from ℂ² to ℂ³, in terms of the Fitting ideals associated with the discriminant.

In this article we consider map germs from (ℂ^n+m, 0) to (ℂ^m, 0), and obtain results to count the number of isolated singularities by means of the dimension of some associated algebras to the Fitting ideals. First in Corollary 4.5 we provide a way to compute the total sum of these singularities. In Proposition 4.9, for m = 3 we show how to compute the number of ordinary triple points. In Corollary 4.10 and with f of co-rank one, we show a way to compute the number of points formed by the intersection between a germ of a cuspidal edge and a germ of a plane.

Furthermore, we show in some examples how to calculate the number of isolated singularities using these results.

These are the notes for lectures given at the Sanya winter school in complex analysis and geometry in January 2016. In Sec. 1, we review the meaning of Ricci curvature of Kähler metrics and introduce the problem of finding Kähler–Einstein metrics. In Sec. 2, we describe the formal picture that leads to the notion of K-stability of Fano manifolds, which is an algebro-geometric criterion for the existence of a Kähler–Einstein metric, by the recent result of Chen–Donaldson–Sun. In Sec. 3, we discuss algebraic structure on Gromov–Hausdorff limits, which is a key ingredient in the proof of the Kähler–Einstein result. In Sec. 4, we give a brief survey of the more recent work on tangent cones of singular Kähler–Einstein metrics arising from Gromov–Hausdorff limits, and the connections with algebraic geometry.

Let $X$ be an elliptic surface over $P^1$ with $\kappa (X)=1$, and $M=M(c_2)$ be the moduli scheme of rank-two stable sheaves $E$ on $X$ with $(c_1(E),c_2(E))=(0,c_2)$ in $Pic(X)\times \mathbb{Z}$. We look into defining equations of $M$ at its singularity $E$, partly because if $M$ admits only canonical singularities, then the Kodaira dimension $\kappa (M)$ can be calculated. We show the following:

• ^(A)$E$ is at worst canonical singularity of $M$ if the restriction of $E_{\eta }$ to the generic fiber of $X$ has no rank-one subsheaf, and if the number of multiple fibers of $X$ is a few.
• ^(B)We obtain that $\kappa (M)=\{1+dim(M)\}/2$ and the Iitaka program of $M$ can be described in purely moduli-theoretic way for $c_2\gg 0$, when $\chi ({{𝒪}}_X)=1$, $X$ has just two multiple fibers, and one of its multiplicities equals $2$.
• ^(C)On the other hand, when $E_{\eta }$ has a rank-one subsheaf, it may be insufficient to look at only the degree-two part of defining equations to judge whether $E$ is at worst canonical singularity or not.

Differential geometry, especially the use of curvature, plays a central role in modern Hodge theory. The vector bundles that occur in the theory (Hodge bundles) have metrics given by the polarizations of the Hodge structures, and the sign and singularity properties of the resulting curvatures have far reaching implications in the geometry of families of algebraic varieties. A special property of the curvatures is that they are $1^{\mathrm{\text{st}}}$ order invariants expressed in terms of the norms of algebro-geometric bundle mappings. This partly expository paper will explain some of the positivity and singularity properties of the curvature invariants that arise in the Hodge theory with special emphasis on the norm property.

A simple two-dimensional system is introduced which suggests a qualitative dynamical relationship between (1) stock market prices in the presence of nonlinear trend-followers and nonlinear value investors, (2) the world human population with a competition between a population-dependent growth rate and a nonlinear dependence on a finite carrying capacity and (3) the failure of materials subjected to a time-varying stress with a competition between positive geometrical feedback on the damage variable and nonlinear healing. Our model keeps three key ingredients (inertia, nonlinear positive and negative feedbacks) that compete to give rise to singularities in finite time decorated by accelerating oscillations.

We describe domain walls that live on A₂ and A₃ singularities. The walls are BPS if the singularity is resolved and non-BPS if it is deformed and fibered. We show that these domain walls may interpolate between vacua that support monopoles and/or vortices.

Cosmological models with time-dependent $\mathrm{\Lambda }$ (read as $\mathrm{\Lambda }(t)$) have been investigated widely in the literature. Models that solve background dynamics analytically are of special interest. Additionally, the allowance of past or future singularities at finite cosmic time in a specific model signals for a generic test on its viabilities with the current observations. Following these, in this work we consider a variety of $\mathrm{\Lambda }(t)$ models focusing on their evolutions and singular behavior. We found that a series of models in this class can be exactly solved when the background universe is described by a spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) line element. The solutions in terms of the scale factor of the FLRW universe offer different universe models, such as power-law expansion, oscillating, and the singularity free universe. However, we also noticed that a large number of the models in this series permit past or future cosmological singularities at finite cosmic time. At last we close the work with a note that the avoidance of future singularities is possible for certain models under some specific restrictions.

We consider a very general scenario of our universe where its geometry is characterized by the Finslerian structure on the underlying spacetime manifold, a generalization of the Riemannian geometry. Now considering a general energy–momentum tensor for matter sector, we derive the gravitational field equations in such spacetime. Further, to depict the cosmological dynamics in such spacetime proposing an interesting equation of state identified by a sole parameter $\gamma$ which for isotropic limit is simply the barotropic equation of state $p=(\gamma -1)\rho$ ($\gamma \in ℝ$ being the barotropic index), we solve the background dynamics. The dynamics offers several possibilities depending on this sole parameter as follows: (i) only an exponential expansion, or (ii) a finite time past singularity (big bang) with late accelerating phase, or (iii) a nonsingular universe exhibiting an accelerating scenario at late time which finally predicts a big rip type singularity. We also discuss several energy conditions and the possibility of cosmic bounce. Finally, we establish the first law of thermodynamics in such spacetime.

We demonstrate how the nonrelativistic high energy asymptotics for the photoionization cross sections of systems bound by a central field can be obtained without solving the wave equations. The earlier analysis carried out for s bound states is extended for p states. We show that the physically expected asymptotics for ionization of fullerenes is not reproduced by the model potentials employed nowadays.

This work deals with an exhaustive study of bouncing cosmology in the background of homogeneous and isotropic Friedmann–Lemaître–Robertson–Walker spacetime. The geometry of the bouncing point has been studied extensively and used as a tool to classify the models from the point of view of cosmology. Raychaudhuri equation (RE) has been furnished in these models to classify the bouncing point as regular point or singular point. Behavior of time-like geodesic congruence in the neighborhood of the bouncing point has been discussed using the Focusing Theorem which follows as a consequence of the RE. An analogy of the RE with the evolution equation for a linear harmonic oscillator has been made and an oscillatory bouncing model has been discussed in this context.

By confining the traveling trajectories of geometrical particles to the null de Sitter space, we describe the trajectory as a spacelike framed surface in the semi-Euclidean space, whereby the Legendrian duality theory and contact manifolds theory, their geometric properties of these spacelike framed surfaces and their singular dual surfaces are investigated in detail. It is of great significance to explore the classifications of singularities of spacelike framed surfaces. We further demonstrated that these spacelike framed surfaces can be interpreted as one-parameter family of framed curves and established the relations between them.

Key issues and essential features of classical and quantum strings in gravitational plane waves, shock waves and space–time singularities are synthetically understood. This includes the string mass and mode number excitations, energy–momentum tensor, scattering amplitudes, vacuum polarization and wave-string polarization effect. The role of the real pole singularities characteristic of the tree level string spectrum (real mass resonances) and that of the space–time singularities is clearly exhibited. This throws light on the issue of singularities in string theory which can be thus classified and fully physically characterized in two different sets: strong singularities (poles of order ≥ 2, and black holes) where the string motion is collective and nonoscillating in time, outgoing states and scattering sector do not appear, the string does not cross the singularities; and weak singularities (poles of order < 2, (Dirac δ belongs to this class) and conic/orbifold singularities) where the whole string motion is oscillatory in time, outgoing and scattering states exist, and the string crosses the singularities.

Common features of strings in singular wave backgrounds and in inflationary backgrounds are explicitly exhibited.

The string dynamics and the scattering/excitation through the singularities (whatever their kind: strong or weak) is fully physically consistent and meaningful.