Linear Pfaffian Research Articles

Linear Pfaffian Systems and Classical Solutions of Triangular Schlesinger Equations

In this paper, by classical solutions we mean solutions to Fuchsian type meromorphic linear integrable Pfaffian systems dy = Ωy on the complex linear spaces ℂn, n ≥ 1, where y(z) = (y1(z),..., yn(z)T ∈ ℂn is a column vector and Ω is a meromorphic matrix differential 1-form such that Ω = ∑1≤i<j≤nJij (β)(zi − zj)−1 d(zi − zj), with constant matrix coefficients Jij(β) depending on complex parameters β = (β1,..., β1). Under some constraints on the constant matrix coefficients Jij (β), the solution components yi(z), 1 ≤ i ≤ n, can be expressed as integrals of products of powers of linear functions; i.e., they are generalizations of the integral representation of the classical hypergeometric function F(z, a, b, c). Moreover, under some additional constraints on the parameters β, the components of the solutions are hyperelliptic, superelliptic, or polynomial functions. We describe such constraints on the coefficients Jij(β) of Fuchsian type systems, as well as describe constraints on the sets of matrices (B1(z),...,Bn(z)) for which the nonlinear Schlesinger equations dBi(z) = $$-\sum{_{j=1, j\neq i}^n}$$[Bi(z),Bj(z)](zi − zj)−1 d(zi − zj) reduce to linear integrable Pfaffian systems of the type described above and have solutions of the indicated type.

Integral Solutions to Schlesinger Equations

It is shown that Schlesinger equations for isomonodromic deformations of Fuchsian systems of order p on the Riemann spheres with upper triangular monodromy are reduced to multidimensional linear homogeneous (p = 2) and inhomogeneous (≥ 3) Pfaffian systems. For components of the solutions to the multidimensional linear Pfaffian systems (p = 2) we obtain integral representations of hypergeometric type and expressions in quadratures close to the hypergeometric Schlesinger equations describing deformations of upper triangular Fuchsian systems of order p = 3.

On a fundamental system of solutions of a certain hypergeometric equation

We study the linear Pfaffian systems satisfied by a certain class of hypergeometric functions, which includes Gauß’s $${}_2 F_{1}$$ , Thomae’s $${}_L F_{L-1}$$ , and Appell–Lauricella’s $$F_D$$ . In particular, we present a fundamental system of solutions with a characteristic local behavior by means of Euler-type integral representations. We also discuss how they are related to the theory of isomonodromic deformations or Painlevé equations.

Pfaffian representations of cubic surfaces

Let K be a field of characteristic zero. We describe an algorithm which requires a homogeneous polynomial F of degree three in K[x_0,x_1,x_2,x_3] and a zero A of F in P^3_K and ensures a linear pfaffian representation of V(F) with entries in K[x_0,x_1,x_2,x_3], under mild assumptions on F and A. We use this result to give an explicit construction of (and to prove the existence of) a linear pfaffian representation of V(F), with entries in K'[x_0,x_1,x_2,x_3], being K' an algebraic extension of K of degree at most six. An explicit example of such a construction is given.

Multidimensional Jordan-Pochhammer systems and their applications

We consider the so-called Jordan-Pochhammer systems, a special class of linear Pfaffian systems of Fuchsian type on complex linear (or projective) spaces. These systems appeared as systems of differential equations for hypergeometric type integrals in which the integrand is a product of powers of linear functions. These systems also arise in some reductions of the Knizhnik-Zamolodchikov equations. The main advantage of these systems is the possibility of presenting a basis in the solution space of such systems in an explicit integral form and, as a consequence, of describing their monodromy representation. The main focus in the paper is placed on the applications of Jordan-Pochhammer systems. We describe the relationship of Jordan-Pochhammer systems to isomonodromic deformations of Fuchsian systems that are described by the Schlesinger equations, as well as to the linearization of the dynamical system of bending spatial polygons. We also describe the application of Jordan-Pochhammer systems to constructing Kohno systems on the Manin-Schechtman configuration spaces.

Pfaffian quartic surfaces and representations of Clifford algebras

Given a general ternary form f=f(x_1,x_2,x_3) of degree 4 over an algebraically closed field of characteristic zero, we use the geometry of K3 surfaces and van den Bergh's correspondence between representations of the generalized Clifford algebra C_f associated to f and Ulrich bundles on the surface X_f:={w^4=f(x_1,x_2,x_3)} \subseteq {P}^3 to construct a positive-dimensional family of 8-dimensional irreducible representations of C_f. The main part of our construction, which is of independent interest, uses recent work of Aprodu-Farkas on Green's Conjecture together with a result of Basili on complete intersection curves in {P}^3 to produce simple Ulrich bundles of rank 2 on a smooth quartic surface X \subseteq {P}^3 with determinant O_X(3). This implies that every smooth quartic surface in {P}^3 is the zerolocus of a linear Pfaffian, strengthening a result of Beauville-Schreyer on general quartic surfaces.

Plane Curves as Pfaffians

Let C be a smooth curve in P given by an equation F = 0 of degree d. In this paper we parametrise all linear pfaffian representations of F by an open subset in the moduli space MC(2, KC). We construct an explicit correspondence between pfaffian representations of C and rank 2 vector bundles on C with canonical determinant and no sections.

HYPERGEOMETRIC SOLUTION OF A CERTAIN POLYNOMIAL HAMILTONIAN SYSTEM OF ISOMONODROMY TYPE

In our previous work, a unified description as polynomial Hamiltonian systems was established for a broad class of the Schlesinger systems including the sixth Painleve equation and Garnier systems. The main purpose of this paper is to present particular solutions of this Hamiltonian system in terms of a certain generalization of Gauss' hypergeometric function. Key ingredients of the argument are the linear Pfaffian system derived from an integral representation of the hypergeometric function (with the aid of twisted de Rham theory) and Lax formalism of the Hamiltonian system.

Elementary transformations of pfaffian representations of plane curves

Let C be a smooth curve in P 2 given by an equation F = 0 of degree d. In this paper we consider elementary transformations of linear pfaffian representations of C. Elementary transformations can be interpreted as actions on a rank 2 vector bundle on C with canonical determinant and no sections, which corresponds to the cokernel of a pfaffian representation. Every two pfaffian representations of C can be bridged by a finite sequence of elementary transformations. Pfaffian representations and elementary transformations are constructed explicitly. For a smooth quartic, applications to Aronhold bundles and theta characteristics are given.

On general surfaces defined by an almost linear pfaffian

We calculate the integers d such that a general surface Xd in \({\mathbb{P}^{3}}\) of degree d contains an arithmetically Gorenstein set of points with a linear syzygy matrix of size 2α + 1. This condition is equivalent to Xd being defined by the pfaffian of a skew-symmetric matrix whose entries are linear except possibly a row and a column. We prove that this takes place for all d ≥ α + 1 if α ≤ 10. Conversely, for α ≥ 11, we show that the condition holds if and only if d is contained in the interval [α + 1, 15].

Tableaux over Lie algebras, integrable systems, and classical surface theory

Starting from suitable tableaux over finite dimensional Lie algebras, we provide a scheme for producing involutive linear Pfaffian systems related to various classes of submanifolds in homogeneous spaces which constitute integrable systems. These include isothermic surfaces, Willmore surfaces, and other classical soliton surfaces. Completely integrable equations such as the G/G(o)-system of Terng and the curved flat system of Ferus-Pedit may be obtained as special cases of this construction. Some classes of surfaces in projective differential geometry whose Gauss-Codazzi equations are associated with tableaux over sl(4, R) are discussed

A linear Pfaffian system at an irregular singularity

A linear Pfaffian system at an irregular singularity

A reduction theorem for a linear Pfaffian system with regular singular points

A reduction theorem for a linear Pfaffian system with regular singular points