Miniversal Deformation Research Articles

On deformation spaces of toric singularities and on singularities of K-moduli of Fano varieties

Firstly, we see that the bases of the miniversal deformations of isolated Q \mathbb {Q} -Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding dimension ≤ 2 \leq 2 which are the bases of the miniversal deformations of isolated Q \mathbb {Q} -Gorenstein toric singularities. Secondly, we show that the deformation spaces of isolated Gorenstein toric 3 3 -fold singularities appear, in a weak sense, as singularities of the K-moduli stack of K-semistable Fano varieties of every dimension ≥ 3 \geq 3 . As a consequence, we prove that the number of local branches of the K-moduli stack of K-semistable Fano varieties and of the K-moduli space of K-polystable Fano varieties is unbounded in each dimension ≥ 3 \geq 3 .

Perturbation theory of matrix pencils through miniversal deformations

We give new proofs of several known results about perturbations of matrix pencils. In particular, we give a direct and constructive proof of Andrzej Pokrzywa's theorem (1983), in which the closure of the orbit of each Kronecker canonical matrix pencil is described in terms of inequalities for invariants of matrix pencils. A more abstract description is given by Klaus Bongartz (1996) by methods of representation theory. We formulate and prove Pokrzywa's theorem in terms of successive replacements of direct summands in a Kronecker canonical pencil.First we show that it is sufficient to prove Pokrzywa's theorem in two cases: for matrices under similarity and for each matrix pencil P−λQ that is a direct sum of two indecomposable pencils. Then we calculate the Kronecker canonical form of pencils that are close to P−λQ. In fact, the Kronecker canonical form is calculated for only those pencils that belong to a miniversal deformation of P−λQ. This is sufficient since all pencils in a neighborhood of P−λQ are reduced to them by a smooth strict equivalence transformation.

Extended r -Spin Theory and the Mirror Symmetry for the A r − 1 -Singularity

By a famous result of K. Saito, the parameter space of the miniversal deformation of the $A_{r-1}$-singularity carries a Frobenius manifold structure. The Landau-Ginzburg mirror symmetry says that, in the flat coordinates, the potential of this Frobenius manifold is equal to the generating series of certain integrals over the moduli space of $r$-spin curves. In this paper we show that the parameters of the miniversal deformation, considered as functions of the flat coordinates, also have a simple geometric interpretation using the extended $r$-spin theory, first considered by T. J. Jarvis, T. Kimura and A. Vaintrob, and studied in a recent paper of E. Clader, R. J. Tessler and the author. We prove a similar result for the singularity $D_4$ and present conjectures for the singularities $E_6$ and $E_8$.

Seiberg–Witten Differential via Primitive Forms

Three-fold quasi-homogeneous isolated rational singularity is argued to define a four dimensional $\mathcal{N}=2$ SCFT. The Seiberg-Witten geometry is built on the mini-versal deformation of the singularity. We argue in this paper that the corresponding Seiberg-Witten differential is given by the Gelfand-Leray form of K. Saito's primitive form. Our result also extends the Seiberg-Witten solution to include irrelevant deformations.

On good deformations of <inline-formula><tex-math id="M1">\begin{document}$ A_m $\end{document}</tex-math></inline-formula>-singularities

The standard way to study a compound singularity is to decompose it into the simpler ones using either blow up techniques or appropriate deformations. Among deformations, one distinguishes between miniversal deformations (related to deformations of a basis of the local algebra of singularity) and good deformations (one-parameter deformations with simple singularities coalescing into a multiple one). In concrete settings, explicit construction of a good deformation is an art rather than a science. In this paper, we discuss some cases important from the application viewpoint when explicit good deformations can be constructed and effectively used. Our applications include: (a) an $ n $-dimensional Euler-Jacobi formula with simple and double roots, and (b) a simple approach to the known classification of phase portraits of planar differential systems around linearly non-zero equilibrium.

The formal Kuranishi parameterization via the universal homological perturbation theory solution of the deformation equation

Abstract Using homological perturbation theory, we develop a formal version of the miniversal deformation associated with a deformation problem controlled by a differential graded Lie algebra over a field of characteristic zero. Our approach includes a formal version of the Kuranishi method in the theory of deformations of complex manifolds.

Perturbation analysis of a matrix differential equation ẋ = ABx

Abstract Two complex matrix pairs (A, B) and (A′, B′) are contragrediently equivalent if there are nonsingular S and R such that (A′, B′) = (S −1 AR, R −1 BS). M.I. García-Planas and V.V. Sergeichuk (1999) constructed a miniversal deformation of a canonical pair (A, B) for contragredient equivalence; that is, a simple normal form to which all matrix pairs (A + A͠, B + B͠) close to (A, B) can be reduced by contragredient equivalence transformations that smoothly depend on the entries of A͠ and B͠. Each perturbation (A͠, B͠) of (A, B) defines the first order induced perturbation AB͠ + A͠B of the matrix AB, which is the first order summand in the product (A + A͠)(B + B͠) = AB + AB͠ + A͠B + A͠B͠. We find all canonical matrix pairs (A, B), for which the first order induced perturbations AB͠ + A͠B are nonzero for all nonzero perturbations in the normal form of García-Planas and Sergeichuk. This problem arises in the theory of matrix differential equations ẋ = Cx, whose product of two matrices: C = AB; using the substitution x = Sy, one can reduce C by similarity transformations S −1 CS and (A, B) by contragredient equivalence transformations (S −1 AR, R −1 BS).

Miniversal deformations of pairs of symmetric matrices under congruence

For each pair of complex symmetric matrices (A,B) we provide a normal form with a minimal number of independent parameters, to which all pairs of complex symmetric matrices (A˜,B˜), close to (A,B) can be reduced by congruence transformation that smoothly depends on the entries of A˜ and B˜. Such a normal form is called a miniversal deformation of (A,B) under congruence. A number of independent parameters in the miniversal deformation of a symmetric matrix pencil is equal to the codimension of the congruence orbit of this symmetric matrix pencil and is computed too. We also provide an upper bound on the distance from (A,B) to its miniversal deformation.

Finite determinacy of matrices over local rings. Tangent modules to the miniversal deformation for R-linear group actions

We consider matrices with entries in a local ring, Matm×n(R). Fix a group action, G⥁Matm×n(R), and a subset of allowed deformations, Σ⊆Matm×n(R). The standard question in Singularity Theory is the finite-(Σ,G)-determinacy of matrices. Finite determinacy implies algebraizability and is equivalent to a stronger notion: stable algebraizability.In our previous work this determinacy question was reduced to the study of the tangent spaces T(Σ,A), T(GA,A), and their quotient, the tangent module to the miniversal deformation, ▪. In particular, the order of determinacy is controlled by the annihilator of this tangent module, ann(T(Σ,G,A)1).In this work we study this tangent module for the group action GL(m,R)×GL(n,R)⥁Matm×n(R) and various natural subgroups of it.We obtain ready-to-use criteria of determinacy for deformations of (embedded) modules, (skew-)symmetric forms, filtered modules, filtered morphisms of filtered modules, chains of modules and others.

On the Moduli Space of Pointed Algebraic Curves of Low Genus III ---Positive Characteristic---

In his classical work, Pinkham discovered a beautiful theorem on the moduli space of pointed algebraic curves with a fixed Weierstrass gap sequence at the marked point. Namely, the complement of a Weierstrass gap sequence in the set of non-negative integers is a numerical semigroup, and he described such a moduli space in terms of the negative part of the miniversal deformation space of the monomial curve of this semigroup. Unfortunately, his theorem holds only in characteristic 0 and does not hold in positive characteristic in general. In this paper, we will study his theorem in positive characteristic, and give a fairly sharp condition for his theorem to hold in positive characteristic up to genus 4. As an application, we present a complete analysis of his theorem in positive characteristic in the low genus case.

Neighborhood radius estimation for Arnold's miniversal deformations of complex and p-adic matrices

V.I. Arnold (1971) constructed a simple normal form to which all complex matrices B in a neighborhood U of a given square matrix A can be reduced by similarity transformations that smoothly depend on the entries of B. We calculate the radius of the neighborhood U. A.A. Mailybaev (1999, 2001) constructed a reducing similarity transformation in the form of Taylor series; we construct this transformation by another method. We extend Arnold's normal form to matrices over the field Qp of p-adic numbers and the field F((T)) of Laurent series over a field F.

Miniversal deformations of pairs of skew-symmetric matrices under congruence

Miniversal deformations for pairs of skew-symmetric matrices under congruence are constructed. To be precise, for each such a pair (A,B) we provide a normal form with a minimal number of independent parameters to which all pairs of skew-symmetric matrices (A˜,B˜), close to (A,B) can be reduced by congruence transformation which smoothly depends on the entries of the matrices in the pair (A˜,B˜). An upper bound on the distance from such a miniversal deformation to (A,B) is derived too. We also present an example of using miniversal deformations for analyzing changes in the canonical structure information (i.e. eigenvalues and minimal indices) of skew-symmetric matrix pairs under perturbations.

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 $[W / {\rm Aut}(C)]$.

Holomorphic Transformation to a Miniversal Deformation not Always Exists Under *Congruence

In 1971, Arnold constructed miniversal deformations of square complex matrices under the similarity transformation. Similar miniversal deformations were constructed for matrices under congruence and under *congruence. For matrices under similarity and under congruence, the holomorphic transformations to their miniversal deformations always exist. We prove that this is not true for matrices under *congruence.

Triality, characteristic classes, $$D_4$$ D 4 and $$G_2$$ G 2 singularities

We recall the construction of triality automorphism of \(\mathfrak {so}(8)\) given by E. Cartan and we give a matrix representation for the real form \(\mathfrak {so}(4,4)\). We compute the induced results on the characteristic classes. Paralelly we study the triality automorphism of the singularity \(D_4\) (in Arnolds classification of smooth functions) and its miniversal deformation. The similarity with Lie theory leads us to a definition of \(G_2\) singularity.

Singularity of type D4 arising from four-qubit systems

An intriguing correspondence between four-qubit systems and simple singularity of type D4 is established. We first consider the algebraic variety X of separable states within the projective Hilbert space . Then, cutting X with a specific hyperplane H, we prove that the X-hypersurface, defined from the section X∩H⊂X, has an isolated singularity of type D4; it is also shown that this is the ‘worst-possible’ isolated singularity that one can obtain by this construction. Moreover, it is demonstrated that this correspondence admits a dual version, by proving that the equation of the dual variety of X, which is nothing but the Cayley hyperdeterminant of type 2 × 2 × 2 × 2, can be expressed in terms of the stochastic local operation and classical communication-invariant polynomials as the discriminant of the miniversal deformation of the D4-singularity. As a consequence of this correspondence one obtains a finer-grained classification of entanglement classes of four-qubit systems.

Miniversal deformations of matrices under *congruence and reducing transformations

Arnold (1971) [1] constructed a miniversal deformation of a square complex matrix under similarity; that is, a simple normal form to which not only a given square matrix A but all matrices B close to it can be reduced by similarity transformations that smoothly depend on the entries of B. We give miniversal deformations of matrices of sesquilinear forms; that is, of square complex matrices under *congruence, and construct an analytic reducing transformation to a miniversal deformation. Analogous results for matrices under congruence were obtained by Dmytryshyn, Futorny, and Sergeichuk (2012) [11].

Analyticity of the total ancestor potential in singularity theory

K. Saito's theory of primitive forms gives a natural semi-simple Frobenius manifold structure on the space of miniversal deformations of an isolated singularity. On the other hand, Givental introduced the notion of a total ancestor potential for every semi-simple point of a Frobenius manifold and conjectured that in the settings of singularity theory his definition extends analytically to non-semisimple points as well. In this paper we prove Givental's conjecture by using the Eynard–Orantin recursion.

An informal introduction to perturbations of matrices determined up to similarity or congruence

The reductions of a square complex matrix A to its canon- ical forms under transformations of similarity, congruence, or *con- gruence are unstable operations: these canonical forms and reduction transformations depend discontinuously on the entries of A. We sur- vey results about their behavior under perturbations of A and about normal forms of all matrices A + E in a neighborhood of A with re- spect to similarity, congruence, or *congruence. These normal forms are called miniversal deformations of A; they are not uniquely deter- mined by A + E, but they are simple and depend continuously on the entries of E.

Moduli spaces of hyperelliptic curves with A and D singularities

We introduce moduli spaces of quasi-admissible hyperelliptic covers with at worst A and D singularities. The stability conditions for these moduli spaces depend on two rational parameters describing allowable singularities. For the extreme values of the parameters, we obtain the stacks of stable limits of \(A_n\) and \(D_n\) singularities, and the quotients of the miniversal deformation spaces of these singularities by natural \(\mathbb G _m\)-actions. We interpret the intermediate spaces as log canonical models of the stacks of stable limits of \(A_n\) and \(D_n\) singularities.