First-order Deformations Research Articles

Cohomology of Associative H -Pseudoalgebras

We define the cohomology of associative [Formula: see text]-pseudoalgebras, and we show that it describes module extensions, abelian pseudoalgebra extensions, and pseudoalgebra first-order deformations. The same results for the special case of associative conformal algebras are also described in details.

Deformation theory of orthogonal and symplectic sheaves

We show that the deformation functor of an orthogonal (resp. symplectic) sheaf over a smooth projective scheme admits a miniversal pro-family, identifying its space of first-order deformations with the first hypercohomology space of a complex which is naturally constructed out of the orthogonal (resp. symplectic) sheaf. We also provide an obstruction theory of these objects whose target is the second hypercohomology space of this complex.

Rigid toric matrix Schubert varieties

Fulton proves that the matrix Schubert variety overline{X_{pi }} cong Y_{pi } times mathbb {C}^q can be defined via certain rank conditions encoded in the Rothe diagram of pi in S_N. In the case where Y_{pi }:={{,textrm{TV},}}(sigma _{pi }) is toric (with respect to a (mathbb {C}^*)^{2N-1} action), we show that it can be described as a toric (edge) ideal of a bipartite graph G^{pi }. We characterize the lower dimensional faces of the associated so-called edge cone sigma _{pi } explicitly in terms of subgraphs of G^{pi } and present a combinatorial study for the first-order deformations of Y_{pi }. We prove that Y_{pi } is rigid if and only if the three-dimensional faces of sigma _{pi } are all simplicial. Moreover, we reformulate this result in terms of the Rothe diagram of pi .

Scalar curvature and deformations of complex structures

Abstract We study a system of equations on a compact complex manifold, that couples the scalar curvature of a Kähler metric with a spectral function of a first-order deformation of the complex structure. The system comes from an infinite-dimensional Kähler reduction, which is a hyperkähler reduction for a particular choice of the spectral function. The main tool for studying the system is a flat connection on the space of first-order deformations of the complex structure, that allows to obtain a formal complexification of the moment map equations. Using this connection, we describe a variational characterization of the equations, a Futaki invariant for the system, and a generalization of K-stability that is conjectured to characterize the existence of solutions.

Rational pullbacks of toric foliations

Abstract This article is dedicated to the study of singular codimension-one foliations ℱ {\mathcal{F}} on a simplicial complete toric variety X and their pullbacks by dominant rational maps φ : ℙ n ⇢ X {\varphi:\mathbb{P}^{n}\dashrightarrow X} . First, we describe the singularities of ℱ {\mathcal{F}} and φ * ⁢ ℱ {\varphi^{*}\mathcal{F}} for a generic pair ( φ , ℱ ) {(\varphi,\mathcal{F})} . Then we show that the first-order deformations of φ * ⁢ ℱ {\varphi^{*}\mathcal{F}} arising from first-order unfoldings are the families of the form φ ε * ⁢ ℱ {\varphi_{\varepsilon}^{*}\mathcal{F}} , where φ ε {\varphi_{\varepsilon}} is a perturbation of φ. We also prove that the deformations of the form φ * ⁢ ℱ ε {\varphi^{*}\mathcal{F}_{\varepsilon}} consist exactly of the families which are tangent to the fibers of φ. In order to do so, we state some results of independent interest regarding the Kupka singularities of these foliations.

ДЕФОРМАЦІЇ ПОВЕРХОНЬ ЗІ СТАЦІОНАРНИМ ТЕНЗОРОМ РІЧЧІ

In this paper, we consider infinitesimal (n. m.) first-order deformations of single-connected regular surfaces in three-dimensional Euclidean space. The search for the vector field of this deformation is generally reduced to the study and solution of a system of four equations (among them there are differential equations) with respect to seven unknown functions. To avoid uncertainty, the following restriction is imposed on a given surface: the Ricci tensor is stored (mainly) on the surface. A mathematical model of the problem is created: a system of seven equations with respect to seven unknown functions. Its mechanical content is established. It is shown that each solution of the obtained system of equations will determine the field of displacement n. m. deformation of the first order of the surface of nonzero Gaussian curvature, which will be an unambiguous function (up to a constant vector). It is proved that each regular surface of nonzero Gaussian and mean curvatures allows first-order n. m. deformation with a stationary Ricci tensor. The tensor fields are found explicitly and depend on two functions, which are the solution of a linear inhomogeneous second-order differential equation with partial derivatives. The class of rigid surfaces in relation to the specified n. m. deformations. Assuming that one of the functions is predetermined, the obtained differential equation in the General case will be a inhomogeneous differential Weingarten equation, and an equation of elliptical type. The geometric and mechanical meaning of the function that is the solution of this equation is found. The following result was obtained: any surface of positive Gaussian and nonzero mean curvatures admits n. m of first-order deformation with a stationary Ricci tensor in the region of a rather small degree. Tensor fields will be represented by a predefined function and some arbitrary regular functions. Considering the Dirichlet problem, it is proved that the simply connected regular surface of a positive Gaussian and nonzero mean curvatures under a certain boundary condition admits a single first-order deformation with a stationary Ricci tensor. The strain tensors are uniquely defined.

Deformations of nonsingular Poisson varieties and Poisson invertible sheaves

We study deformations of nonsingular Poisson varieties and Poisson invertible sheaves, which extend the flat deformation theory of nonsingular varieties and invertible sheaves. In the Appendix, we study deformations of Poisson vector bundles. We identify first-order deformations and obstructions.

Marginal deformations of 3d mathcal{N}=2 CFTs from AdS4 backgrounds in generalised geometry

We study exactly marginal deformations of 3d mathcal{N}=2 CFTs dual to AdS4 solutions in eleven-dimensional supergravity using generalised geometry. Focussing on Sasaki-Einstein backgrounds, we find that marginal deformations correspond to turning on a four-form flux on the internal space at first order. Viewing this as the deformation of a generalised structure, we derive a general expression for the four-form flux in terms of a holomorphic function. We discuss the explicit examples of S7, Q1,1,1 and M1,1,1 and, using an obstruction analysis, find the conditions for the first-order deformations to extend all orders, thus identifying which marginal deformations are exactly marginal. We also show how the all-orders γ-deformation of Lunin and Maldacena can be encoded as a tri-vector deformation in generalised geometry and outline how to recover the supergravity solution from the generalised metric.

A simple construction of associative deformations

We propose a simple approach to formal deformations of associative algebras. It exploits the machinery of multiplicative coresolutions of an associative algebra A in the category of A-bimodules. Specifically, we show that certain first-order deformations of A extend to all orders and we derive explicit recurrent formulas determining this extension. In physical terms, this may be regarded as the deformation quantization of noncommutative Poisson structures on A.

The dualizing sheaf on first-order deformations of toric surface singularities

Abstract We explicitly describe infinitesimal deformations of cyclic quotient singularities that satisfy one of the deformation conditions introduced by Wahl, Kollár–Shepherd-Barron (KSB) and Viehweg. The conclusion is that in many cases these three notions are different from each other. In particular, we see that while the KSB and the Viehweg versions of the moduli space of surfaces of general type have the same underlying reduced subscheme, their infinitesimal structures are different.

Non-commutative deformations and quasi-coherent modules

We identify a class of quasi-compact semi-separated (qcss) twisted presheaves of algebras $$\mathcal {A}$$ for which well-behaved Grothendieck abelian categories of quasi-coherent modules $$\mathsf {Qch} (\mathcal {A})$$ are defined. This class is stable under algebraic deformation, giving rise to a 1–1 correspondence between algebraic deformations of $$\mathcal {A}$$ and abelian deformations of $$\mathsf {Qch} (\mathcal {A})$$ . For a qcss presheaf $$\mathcal {A}$$ , we use the Gerstenhaber–Schack (GS) complex to explicitly parameterize the first-order deformations. For a twisted presheaf $$\mathcal {A}$$ with central twists, we descibe an alternative category $$\mathsf {QPr} (\mathcal {A})$$ of quasi-coherent presheaves which is equivalent to $$\mathsf {Qch} (\mathcal {A})$$ , leading to an alternative, equivalent association of abelian deformations to GS cocycles of qcss presheaves of commutative algebras. Our construction applies to the restriction $$\mathcal {O}$$ of the structure sheaf of a scheme X to a finite semi-separating open affine cover (for which we have $$\mathsf {Qch} (\mathcal {O}) \cong \mathsf {Qch} (X)$$ ). Under a natural identification of GS cohomology of $$\mathcal {O}$$ and Hochschild cohomology of X, our construction is shown to be equivalent to Toda’s construction from Toda (J Differ Geom 81(1):197–224, 2009) in the smooth case.

The Poincaré algebras p(1,1) and p(1, 2): realizations and deformations

All inequivalent realizations of the Poincaré algebras p(1,1) and p(1, 2) acting in spaces of not more than three variables are constructed. First-order deformations of p(1,1) and p(1,2) are proposed and the generic realizations for the initial and deformed Poincaré algebras are presented.

Extensions de représentations de de Rham et vecteurs localement algébriques

Let${\rm\Pi}$be an irreducible unitary completion of a locally algebraic$\text{GL}_{2}(\mathbf{Q}_{p})$-representation. We describe those first-order deformations of${\rm\Pi}$which are themselves completions of a locally algebraic representation. This answers a question of Paškūnas and has direct applications to the Breuil–Mézard conjecture.

Combined influences of shear deformation, rotary inertia and heterogeneity on the frequencies of cross-ply laminated orthotropic cylindrical shells

The non-dimensional frequencies for symmetric and anti-symmetric cross-ply laminated heterogeneous composite circular cylindrical shells are analyzed by taking into account the effects of first-order deformations such as transverse shear deformations and rotary inertia. By using the Donnell-type shell theory, a set of fundamental dynamic equations of laminated circular cylindrical shells made of heterogeneous orthotropic materials is derived through Hamilton’s principle. The basic equations are reduced to the six-order algebraic equation. One of the lowest positive roots of the algebraic equation represents the fundamental frequency. Attention is focused on the case of cross-ply laminated heterogeneous orthotropic cylindrical shells, from which solution for homogenous and heterogeneous orthotropic monolayer cylindrical shells follows based on classical shell theory (CST) and shear deformation theory (SDT), as a special case. Moreover, further detailed numerical results dealing with non-dimensional frequencies and corresponding mode shapes of laminated heterogeneous cylindrical shells having symmetric or anti-symmetric cross-ply lay-up are discussed. Furthermore, some comparisons are made to show the reliability and accuracy of the study.

On metastable vacua and the warped deformed conifold: analytic results

Continuing the programme of constructing the backreacted solution corresponding to smeared anti-D3 branes in the warped deformed conifold, we solve analytically the equations governing the space of first-order deformations around this solution. We express the results in terms of at most three nested integrals. These are the simplest expressions for the space of -invariant deformations, in which the putative solution for smeared anti-D3 branes must live. We also explain why one cannot claim to identify this solution without fully relating the coefficients of the infrared and ultraviolet expansions of the deformation modes. The analytic solution we find is the first step in this direction.

On the simultaneous recognition of identity and expression from BU-3DFE datasets

We propose a new linear model, based on resampling 3D face meshes to convert them to 3D matrices, to recognize identity and expression simultaneously. This contrasts with bilinear models currently used with 3D meshes. The matrices are amenable to algebraic operations for facial data analysis and synthesis. Facial emotion is represented as a linear combination of its identity and expression using principal components extracted from training data as neutral-to-emotion deformations. The linear model is applicable to other mesh data with pose variations after correction using recently available techniques. The proposed approach avoids the problem of correspondence between pairs of person’s neutral and emotion meshes for estimating facial deformations used as features. Identity and expression recognition accuracies, obtained by representing resampled matrices as linear combinations of composite depth-color (gray) PCs, are better than the results in the literature on both simultaneous identity-expression using bilinear models and expression-only recognition using deformable models, facial action codes, distances between pairs of annotated facial points as features and others. The proposed framework can also be used to generate synthetic matrices displaying a wide array of natural and mixed emotions for any chosen identity. A byproduct is the result that second-order deformations as features do not seem to perform as effectively as first-order deformations for identity and expression recognition.

Deformations of rational 𝑇-varieties

We show how to construct certain homogeneous deformations for rational normal varieties with codimension one torus action. This can then be used to construct homogeneous deformations of any toric variety in arbitrary degree. For locally trivial deformations coming from this construction, we calculate the image of the Kodaira-Spencer map. We then show that for a smooth complete toric variety, our homogeneous deformations span the space of first-order deformations.

Infinitesimal Derived Torelli Theorem for K3 Surfaces (with an Appendix by Sukhendu Mehrotra)

We prove that the first-order deformations of two smooth projective K3 surfaces are derived equivalent under a Fourier–Mukai transform if and only if there exists a special isometry of the total cohomology groups of the surfaces which preserves the Mukai pairing, an infinitesimal weight-2 decomposition and the orientation of a positive four-dimensional space. This generalizes the derived version of the Torelli theorem. Along the way we show the compatibility of the actions on Hochschild homology and singular cohomology of any Fourier–Mukai functor.

Cohomology and deformation theory of monoidal 2-categories I

In this paper we define a cohomology theory for an arbitrary K-linear semistrict semigroupal 2-category ( C,⊗) (called for short a Gray semigroup) and show that its first-order (unitary) deformations, up to the suitable notion of equivalence, are in bijection with the elements of the second cohomology group. Fundamental to the construction is a double complex, similar to the Gerstenhaber–Schack double complex for bialgebras, the role of the multiplication and the comultiplication being now played by the composition and the tensor product of 1-morphisms. We also identify the cohomologies describing separately the deformations of the tensor product, the associator and the pentagonator. To obtain the above results, a cohomology theory for an arbitrary K-linear (unitary) pseudofunctor is introduced describing its purely pseudofunctorial deformations, and generalizing Yetter's cohomology for semigroupal functors (in: M. Kapranov, E. Getzler (Eds.), Higher Category Theory, AMS Contemporary Mathematics, Vol. 230, Amer. Math. Soc., Providence, RI, 1998, pp. 117–134). The corresponding higher order obstructions will be considered in detail in a future paper.

Gauge-covariant deformations, symmetries, and free parameters of string theory.

We examine the first-order deformations of the stress tensor of the free bosonic conformal field theory which preserve the two mutually commuting copies of the Virasoro algebra and involve just two derivatives. We find a much richer structure than canonical deformations, and, without using ghosts, exhibit a distinct deformation of the stress tensor for every solution to the linearized covariant equations of motion for the massless modes of the bosonic string. We are thus able to derive explicitly all general coordinate and two-form gauge transformations, about flat space-time, and we argue that all zero modes generate symmetries if we admit auxiliary fields. We also show that, to first order, there exists in addition a finite number of deformations, associated with world-sheet instantons, which may correspond to free parameters of string theory.