Versal Deformations Research Articles

On cohomology and deformations of Jacobi-Jordan algebras

In this paper we study cohomology and deformations of Jacobi-Jordan algebras. We develop their formal deformation theory. In particular, we introduce a method to construct a versal deformation for a given Jacobi-Jordan algebra, which can induce all deformations and is unique on the infinitesimal level. We construct formal 1-parameter deformations of Jacobi-Jordan algebras up to dimension 5 and versal deformations for 3-dimensional Jacobi-Jordan algebras.

Perturbation of Multiagent Linear Systems

The main objective of this note is to explore if, making a small perturbation of an uncontrollable multiagent linear system with a previously interrelationship topology established, a controllable multi-agent system with the same topology can be obtained. Arnold geometric techniques will be used for the objective, and versal deformations will be constructed in the set of equivalent systems.

Proof of the elliptic expansion moonshine conjecture of Căldăraru, He, and Huang

Using predictions in mirror symmetry, Căldăraru, He, and Huang recently formulated a “Moonshine Conjecture at Landau-Ginzburg points” [arXiv:2107.12405, 2021] for Klein’s modular j j -function at j = 0 j=0 and j = 1728. j=1728. The conjecture asserts that the j j -function, when specialized at specific flat coordinates on the moduli spaces of versal deformations of the corresponding CM elliptic curves, yields simple rational functions. We prove this conjecture, and show that these rational functions arise from classical 2 F 1 _2F_1 -hypergeometric inversion formulae for the j j -function.

Algebraic deformations and Fourier–Mukai transforms for Calabi–Yau manifolds

Given a pair of derived-equivalent Calabi–Yau manifolds of dimension more than two, we prove that the derived equivalence can be extended to general fibers of versal deformations. As an application, we give a new proof of the Pfaffian–Grassmannian derived equivalence.

Non-commutative deformations of perverse coherent sheaves and rational curves

We consider non-commutative deformations of sheaves on algebraic varieties. We develop some tools to determine parameter algebras of versal non-commutative deformations for partial simple collections and the structure sheaves of smooth rational curves. We apply them to universal flopping contractions of length 2 2 and higher. We confirm Donovan-Wemyss conjecture in the case of deformations of Laufer’s flops.

A support theorem for\break the Hitchin fibration: The case of GL n and K C

Abstract We compute the supports of the perverse cohomology sheaves of the Hitchin fibration for GL n {\mathrm{GL}_{n}} over the locus of reduced spectral curves. In contrast to the case of meromorphic Higgs fields we find additional supports at the loci of reducible spectral curves. Their contribution to the global cohomology is governed by a finite twist of Hitchin fibrations for Levi subgroups. The corresponding summands give non-trivial contributions to the cohomology of the moduli spaces for every n ≥ 2 {n\geq{2}} . A key ingredient is a restriction result for intersection cohomology sheaves that allows us to compare the fibration to the one defined over versal deformations of spectral curves.

Non-commutative deformations of simple objects in a category of perverse coherent sheaves

We define a category of perverse coherent sheaves as the abelian category corresponding to the category of modules under Bondal–Rickard equivalence which arises from a tilting bundle for a projective morphism. The purpose of this paper is to determine versal non-commutative deformations of simple collections in the categories of perverse coherent sheaves in some cases. In general we prove that the non-commutative structure algebra is recovered as the parameter algebra of the versal non-commutative deformation of the simple collection consisting of all simple objects over a closed point of the base space. In the case where the fiber dimensions are at most 1 and the structure sheaf is relatively acyclic, we determine the versal deformations of some partial simple collections consisting of vanishing simple objects. In particular it is proved that the parameter algebra of the versal non-commutative deformation is isomorphic to its opposite algebra in this case.

Differentiable families of traceless matrix triples

Analysis of spectra of families of sets of matrices verifying certain properties is not simple because phenomena as singularities and bifurcations appear. An excellent tool for the analysis can be making use of versal deformations because of the spectrum of the family coincides with the spectrum of its versal deformation. Disposing of a versal deformation is advantageous since any perturbation of an element can be described up to equivalence by its versal deformation, and it gives the possibility to calculate bifurcation diagrams of families of elements in general position. V.I. Arnold constructed versal deformations, of a differentiable family of square matrices under conjugation and his techniques have been generalised to different cases as to matrix pencils under the strict equivalence, for example. In this paper, we present versal deformations of elements of the Lie algebra consisting of triples of traceless matrices to coefficients on $${\mathbb {F}} ={\mathbb {C}}$$ or $${\mathbb {R}}$$, which are simultaneously diagonalizable. Study families of traceless matrix triples have great interest because the Lie algebra is related to gauge fields because they appear in the Lagrangian describing the dynamics of the field, then they are associated to 1-forms that take values on a certain Lie algebra. It is also of interest to note that triples of traceless matrices have some relevance for supergravity theories. Another application is found when we must give the instanton solution of Yang-Mills field can be presented in an octonion form, and it can be represented by triples of traceless matrices.

In Memoriam: Subhash C. Sinha

In Memoriam: Subhash C. Sinha

Versal deformations of matrix products

For each pair (A, B) of $$m\times n$$ and $$n\times m$$ complex matrices, Garcia-Planas and Sergeichuk (Linear Algebra Appl 302(303):45–61, 1999) constructed a family of pairs $$(A_{\text {gs}},B_{\text {gs}})$$ of a simple form to which all matrix pairs $$(A', B')$$ close to (A, B) can be reduced by transformations $$(S^{-1}A'R,R^{-1}B'S)$$ given by nonsingular matrices $$S=S(A',B')$$ and $$R=R(A',B')$$ that smoothly depend on the entries of $$A'$$ and $$B'$$. We describe all pairs (A, B) of $$m\times n$$ and $$n\times m$$ complex matrices with the following property: all $$m\times m$$ matrices $$M'$$ close to AB can be reduced to the form $$A_{\text {gs}}B_{\text {gs}}$$ by similarity transformations that smoothly depend on the entries of $$M'$$.

Multiplicities of Bifurcation Sets of Pham Singularities

The local multiplicities of the Maxwell sets in the spaces of versal deformations of Pham holomorphic function singularities are calculated. A similar calculation for some other bifurcation sets (generalized Stokes' sets) defined by more complicated relations between the critical values is given. Aplications to the complexity of algorithms enumerating topologically distinct morsifications of complicated real function singularities are discussed.

Flat coordinates for Saito Frobenius manifolds and string theory

It was shown in \cite{DVV} for $2d$ topological Conformal field theory (TCFT) \cite{EY,W} and more recently in \cite{BSZ}-\cite{BB2} for the non-critical String theory \cite{P}-\cite{BAlZ} that a number of models of these two types can be exactly solved using their connection with the Frobenius manifold (FM) structure introduced by Dubrovin\cite{Dub}. More precisely these models are connected with a special case of FMs, so called Saito Frobenius manifolds (SFM)\cite{Saito} (originally called Flat structure together with the Flat coordinate system), which arise on the space of the versal deformations of the isolated Singularities after choosing of a suitabe so-called Primitive form, and which also arises on the quotient spaces by reflection groups. In this paper we explore the connection of the models of TCFT and non-critical String theory with SFM. The crucial point for obtaining an explicit expression for the correlators is finding the flat coordinates of SFMs as functions of the parameters of the deformed singularity. We suggest a direct way to find the flat coordinates, using the integral representation for the solutions of Gauss-Manin system connected with the corresponding SFM for a simple singularity. Also, we address the possible generalization of our approach for the models investigated in \cite{Gep} which are ${SU(N)_k}/({SU(N-1)_{k+1} \times U(1)})$ Kazama-Suzuki theories \cite{KS}.

The Pfeiffer-Lax-Sato type vector field equations and the related integrable versal deformations

The Pfeiffer-Lax-Sato type vector field equations and the related integrable versal deformations

Equivariant versal deformations of semistable curves

We prove that given any $n$-pointed prestable curve $C$ of genus $g$ with linearly reductive automorphism group ${\rm Aut}(C)$, there exists an ${\rm Aut}(C)$-equivariant miniversal deformation of $C$ over an affine variety $W$. In other words, we prove that the algebraic stack $\mathfrak{M}_{g,n}$ parametrizing $n$-pointed prestable curves of genus $g$ has an \'etale neighborhood of $[C]$ isomorphic to the quotient stack $[W / {\rm Aut}(C)]$.

The moduli space of complex 5-dimensional Lie algebras

In this paper, we study the moduli space of all complex 5-dimensional Lie algebras, realizing it as a stratification by orbifolds, which are connected only by jump deformations. The orbifolds are given by the action of finite groups on very simple complex manifolds. Our method of determining the stratification is based on the construction of versal deformations of the Lie algebras, which allow us to identify natural neighborhoods of the elements in the moduli space.

Multiplicities of Bifurcation Sets of Pham Singularities

The local multiplicities of the Maxwell sets in the spaces of versal deformations of Pham holomorphic function singularities are calculated. A similar calculation for some other bifurcation sets (generalized Stokes' sets) defined by more complicated relations between the critical values is given. Aplications to the complexity of algorithms enumerating topologically distinct morsifications of complicated real function singularities are discussed.

The first, second and fourth Painlevé equations on weighted projective spaces

The first, second and fourth Painlevé equations are studied by means of dynamical systems theory and three dimensional weighted projective spaces CP3(p,q,r,s) with suitable weights (p,q,r,s) determined by the Newton diagrams of the equations or the versal deformations of vector fields. Singular normal forms of the equations, a simple proof of the Painlevé property and symplectic atlases of the spaces of initial conditions are given with the aid of the orbifold structure of CP3(p,q,r,s). In particular, for the first Painlevé equation, a well known Painlevé's transformation is geometrically derived, which proves to be the Darboux coordinates of a certain algebraic surface with a holomorphic symplectic form. The affine Weyl group, Dynkin diagram and the Boutroux coordinates are also studied from a view point of the weighted projective space.

Extensions of associative algebras

In this paper, we translate the problem of extending an associative algebra by another associative algebra into the language of codifferentials. The authors have been constructing moduli spaces of algebras and studying their structure by constructing their versal deformations. The codifferential language is very useful for this purpose. The goal of this paper is to express the classical results about extensions in a form which leads to a simpler construction of moduli spaces of low-dimensional algebras.

On near and the nearest correlation matrix

We present an elementary heuristic reasoning based on Arnold's theory of versal deformations in support of a straightforward algorithm for finding a correlation matrix near a given symmetric one.

The Wigner caustic on shell and singularities of odd functions

We study the Wigner caustic on shell of a Lagrangian submanifold L of affine symplectic space. We present the physical motivation for studying singularities of the Wigner caustic on shell and present its mathematical definition in terms of a generating family. Because such a generating family is an odd deformation of an odd function, we study simple singularities in the category of odd functions and their odd versal deformations, applying these results to classify the singularities of the Wigner caustic on shell, interpreting these singularities in terms of the local geometry of L.